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The  Dihner  Library 
of  the  History  of 
Science  and  Technology 

SMITHSONIAN  INSTITUTION  LIBRARIES 


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THE  ELEMENTS 

OF  GHOMHTRIH 

of  the  moll  aunci- 
enc  Philolopher 
EFCLIDE 

ofMeaara, 


Strabo 


[jtralus 


Faithfully  ( now firsl  )  tran* 
fated  into  the  Englife  toung ,  by 
H.  BillingWiy  ,Cuiz.en  of  London, 
Whereanto  are  annexed  certaine 

Scholtes ,  Annotations, and  Indenti¬ 
ons,  of  the  ieff  Mathematici¬ 
ans,  both  of  time  paft ,  and 
in  this  our  age , 


cPoltbt 


j  lljfarcfms 


R-tisssssns^^ss 


\  With  a  very  fir  uitf nil  Preface  made  by  M.  I.  Dee, 
;  jpecifying  the  chief  if1! athematicali  Scieces,  Vehat 

\  thej  are, and  vrherunto  commodious -.where, alfo,  are 

s  difclo  fed  certaine  new  Secrets  Mathematicall 

\  and  McchanicallswitiU  theft  our  dates, greatly  miffed. 


Mtronomia 


Geometria 


'u\mi 


Mu  sica 


Anthmctica 


illL  \\yi(  H Y.1‘  l'Yfi  'Y  ,  a v 

Wu  nV/JSAVi 

00  0  ® 

isi 

ywfyf/ 

Imprinted  at  London  by holm  ID  aye. 


5*  The  T ranikior  to  the  Reader. 


Here  is  (gentle  Trader')  nothing 
( the  wordoffod  oncly fet  apart') 
which  fo  much  beautifieth  and  a- 
dorneth  the  foule^  and  minde^  of 
ma.  as  doth  the  knowledge  0 
andfciences:  as  th 
ye  ofnaturalland  moraU'Ttoi- 
ojophie  .  The  onefetteth  before 
our  eyes.  the^  creatures  offfod. 

:  in  which  as 


both  in 

in  o^glafe.  we  beholde  the  exceding  maiefiie  andwifedome  of 
fjod.  in  adorning  and  beautifying  them  as  we fee :  ingeuingyn - 
to  them fuch  wonder  full  and mani folde proprieties .  and  naturall 
workjnges.  and  that  fodiuerflyand  in  fuck  yarietie :  farther  in 
maintaining  and  conferuing  them  continually  .whereby  to praife 
and  adore  him .  as  by  SIPaule  we  are  taugh  t  *  The  other  tea t- 
cheth  ys  rules  and '  preceptes  of  yertue.how yin  common  life'4-j- 
mongef  men u,  we  ought  to  walke  yprightly  :what  dueties per- 
taine  to  our felues.  whatpertaine  to  the  gouernment  or  good  or** 
der  both  of  an  houfholde.  and  alfo  of  a  citie  or  commonwealth. 
The  reading  lihpwife  ofhifiories.conduceth  not  a  htle.to  the  ad¬ 
orning  of  the  foule  &  minde  of  man  .a fiudie  of  all  men  comen¬ 
ded:  by  it  are feene  and  fnowen  the  artes  and  doinges  of  infinite 
wife  men  gone  before  ys .  In  hifiories  are  contained  infinite  ex¬ 
amples  ofheroicall  yertues  to  be  of  ys followed. and  horrible  ex¬ 
amples  offices  to  be  ofys  efchewed  .  oZAdany  other  artes  alfo 
there  are  which  beautifie  the  minde  of  man:  but  of  all  other  none 
do  more garnifhe  &  beautifie  it.  then  thofe  artes  which  are  cal¬ 
led  Ala  t hematic  a  ll .  Unto  the  knowledge  of  which  no  man  can 
attaine  .without  the  perfeBe  knowledge  md  infiruBion  of  the 
principles  .groundes  .and Elementes  cf Geometric  ,  Hut  per- 
y  83s*  H*  feBly 


$&  TheTranflator  to  the  Reader, 

feBly  to  be  injlruBed  in  them,  requireth  diligent  fludie  and  rest * 
dingof  olde  auncient  authors .  fAmongefl  which, none for  a  be¬ 
ginner  is  to  be  preferred  before  the  mofl  auncient  Fhilofopher 
Euclide  o/TVlegara .  For  of  all  others  he  hath  in  a  truc^  me¬ 
thods  and  infle  order,  gathered  together  whatfoeuer  any  before 
himhad  ofthefe  Element  es  written:  inuenting  alfo  and  adding 
mar\y  t  hinges  of  his  owne :  wherby  he  hath  in  due  forme  accom¬ 
pli  fhed  the  artefrfgeuing  definitions  principles,  &  ground es , 
wherofhe  deduceth  his  F  r  op  o fit  ions  or  conclufions fnfuch  won - 
derfullwife,  that  that  which goeth  before  ,  isofnecefitie  requi¬ 
red  to  theprouf-j  t f that  which follow  eth  .  So  thatwithout  the 
diligent fludie  of  Euclides  Element es, it  is  impofiible  to  attaine 
•vnto  the perfette  knowledge  of  Cfcometrie,  and  confequently  of 
any  of  the  other  Math  ematic all  jciences  .  Wherefore  confide - 
ring  the  want  &  lacfe  offuchgood  authors  hitherto  in  our  Eng- 
hfhe  lounge,  lamenting  alfo  the  negligence,  and  lachp  of  ge alts 
to  their  countrey  inthofe  of  our  nation ,  to  whom  Cfod  hath  geuen 
loth  knowledge,  &  alfo  abilitie  to  tranflate  into  our  tounge,and 
to pub  lifhe  abroad  fuch good  authors, and  booses  ( the  chiefe  in- 
flrumentesofalllearninges')  :  feing  moreouer  that  many  good 
wittes  both  of  gentlemen  and  of  others  of  all  degrees,  much  de- 
firousandjludiousofthefe  artes,  and  feeling  for  them  as  much 
as  they  cm, faring  no  games,  andyetfrufirate  of  their  intent, 
by  no  meancs  attaining  to  that  which  they  feekgs :  I haue^>  for 
their fakes, with fome  charge  &  great  trauaile, faithfully  tran- 
Jlated  into  ourtulgare  touge,(yfet  abroad  in  Frint ,  this  booke 
^Euclide.  Whereunto  I haue  added eafie and plaine decla¬ 
rations  and  examples  by  figures,  of  the  definitions  .  In  which 
bookp  alfoye  fhall  in  due  place  finde  manifolde  additions,  Scho- 
lies.  Annotations, and  Inuentions :  which  I  haue  gathered  out  of 
many  of  the  mofl famous  &  chiefe  Mathematics  s ,  both  of old 
time, and  in  our  age:  as  by  diligent  reading  it  in  courfe,ye fhall 

well 

m 


S-i/The  Tranflater  to  the  Reader, 

Well perceaue .  Fhe fruite  and game  which  I  require  for  theft 
my paines  and trauailefeall be  nothing  els,  but  onely  that  thou 
gentle  reader ,  will  gratefully  accept  the  fame^ :  and  that  thou 
may  eft  thereby  receaue  jome profite:and moreouer  to  excite  and 
fir  re  vp  others  learned \  to  do  the  like >  to  take  paines  in  that 

behalfe .  'By  meanes  wherofour  Bnghfhe  tounge fall  no  lejfe  be 
enriched  with  good  Authors  >  then  are  other  jlraunge  tounge s: 
as  the  Butch ,  French  ,  ftalian  ,  and  Spanifhe  :  in  which 
are  red  all  good  authors  in  a  maner, found  amongef  the  (ftrekes 
or  Latines .  Which  is  the  chief ef  caufe ,  that  amongef  the  do  flo-* 
rijhe  fo  many  cunning  and  fkilfull  men ,  in  the  inuentions  of 
fraunge  and  wonderfull  t hinges,  as  in  thefe  ourdaies 
we fee  there  do  .  Which fruite  andgaine  if  I attaine 
Dntoy  itjhall  encourage  me  hereafter,  in  fuch  like 
fort  to  tranfate ,  and fet  abroad  feme  other 
good  authors,  both  pertaining  to  religion 
(  as  partly  I  haue  already  done  )  and 
alfo  pertaining  to  the  <£\d'athe~ 
maticall  Artes .  Thus  gentle 
reader  farewell* 

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.  'so mow./,1  -as}. '  •  o,  •. '*Avv\\Va\  \mu>vovi!fv.*  o\  ,1\V \ 

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5**>TO  the  vnfained  lovers 

of  truthe  ,  and  conftant  Studentes  of  Noble 

Sciences  O  H  N  DEE  of  London Jhartilj 
wifheth  grace  from  heauen,  and  moft  profpe- 
rous JucceJJe  in  all  their  honejl  attempt es  emd 
exercifes* 

luine  TlatOy  the  great  M after 
of  many  worthy  Philofophers* 
and  the  conftant  auoucher,  and 
pithy  perfwader  of  Vnurn  *  Bo- 
nttm ,  and  Ens :  in  his  Schole  and 
Academic,  fundry  times  (befides 
his  ordinary  Scholers)  was  vifited 
of  a  certaine  kinde  of  men,  allured 
by  the  noble  fame  of  Plato,  and 
the  great  commendation  of  hys 
profound  and  profitable  doctrine* 
But  when  fuch  Hearers,after  long 
harkening  to  him,  perceaued,  that 
the  drift  of  his  difcourfes  ifftied 
out,  to  conclude ,  this  Vnttm ,  Bo* 
num ,and  Ens-,  to  be  Spirituall,Infi- 
nite,  iEternall ,  Omnipotent ,  &c. 
Nothyngbeyngalledged  or  exprefled,How, worldly  goods:  how,  worldly  digni- 
tie:how,health,Stregth  or  luftines  of bodyrnor  yet  the  meanes,how  a  merueilous 
fenfibleand  bodyly  blyfte  and  felicitie  hereafter,  might  beatteyned:  Straightway* 
the  fantafies  of  thofe  hearers,were  dampt:  their  opinion  ofjP/^t^was  dene  chaun- 
ged:yea  his  dodrine  was  by  them  defpifed:and  his  fchole ,  no  more  of  them  vifi- 
ted.Which  thing,his  S  choler,  Arijlotle,  narrowly  cofictering/ounde  the  caufe  ther- 
of,to  be, For  that  they  had  no  forwarnyng  and  information,in  general! ,  whereto  >t> 
his  dodrine  tended.For,fo,might  they  hauc  had  occafion, either  to  haue  forborne 
his  fchole  hauntyng :  (if  they, then, had  mifliked  his  Scope  and  purpofe  )  or  con- 
ftandy  to  haue  continued  therinrto  their  full  fatiftadion  :  if  fuch  his  finallfcope& 
intent ,  had  ben  to  their  defire .  Wherfore*  Arijlotle, after  that,vfed  in  brief,to 
forewarne  his  owne  Scholers  and  hearers ,  both  of  what  matter  >  and  alfo  to  what 
ende,he  tooke  in  hand  to  fpeake,  or  teach .  While  I  confider  the  diuerfe  trades  of ,j* 
thefe  two  excellent  Philofbphers  (  and  am  moft  fure,both,  that  Plato  right  well,  o- 
therwife  could  teach :  and  that  (^Arijlotle  mought  boldely ,  with  his  hearers ,  haue 
dealt  in  like  forte  as  Plato  did)I  am  in  no  little  pang  of  perplexitie :  Bycaufe ,  that, 
which  I  miflike,is  moft  eafy  forme  to  performe  (and  to  haue  Plato  for  my  exaple.) 
And  that, which  I  know  to  be  moft  commendable:  and  (in  this  firft  bringyng,into 
common  handling, the  <_ Antes  CEtathematicall)  to  be  moft  neceftary :  is  full  of  great 
difficultie  and  fundry  daungers.Y et, neither  do  I  think  it  mete, for  fo  ftraunge  mat- 
ter(as  now  is  ment  to  be  publifhed)and  to  fo  ftraunge  an  audience ,  to  be  bluntly, 
at  firft, put  forth, without  a  peculiar  Preface  :  Nor  (Imitatyng  Arijlotle)  well  cart  I 
hope ,  that  accordyng  to  the  amplenes  and  dignitie  of  the  State  CMathematicall ,  I 
am  able, either  playnly  to  prefcribe  the  materiall  boundes  :  or  precifely  to  expreftc 
the  chief  purpofes,  and  moft  wonderfull  applications  therof.  And  though  lam 
fure ,  that  fuch  as  didfhrinke  from  Plato  his  fchole ,  after  they  had  perceiued  his  ft- 


lolin  D^e  his  Mathematical!  Preface. 

ilall  conclufion,  would  in  thefe  thinges  haue  ben  his  moft  diligent  hearers  )  fo  infi¬ 
nitely  mought  their  defires, in  fine  and  at  length  ,  by  our  Artes  Mathematical!  be  fa- 
tifHed)yet,by  this  my  Preface  &  fbrewarnyng ,  Afwell  all fuchgiiay  (to  their  great 
behofe)the  loner,  hither  be  alluredras  alfo  the  Pphagoricalf  and  Platomcall  perfect 
fcholer,and  the  conftant  profound  Philofopher,  with  more  eafeand  fpede,may 
(like  the  Bee,)  gather, hereby,both  wax  and  hony. 

Wherfore, fey  ng  I  finde  great  occafion(for  the  caufes  alleged, and  fafder,  in  re- 
5?  fped  of  my  Art  CMathcmatike  general! )  to  vfe  a  certaine  forewarnyng  and  Preface, 
»  whofe  content  flialbe,that  mighty, moft  plefaunt,and frutefull  Mathematical!,  Tree , 
The  intent  of  with  his  chief  armes  and  fecond(grifted)braunches:  Both, what  euery  one  is ,  and 
this  Preface .  alfo,  what  commodity, in  general!, is  to  be  looked  for,afwelI  of  griff  as  ftockeAnd 
„  forafmuch  as  this  enterprife  is  fo  great,  that, to  this  our  tyme ,  it  neuer  was  (to  my 
,,  knowledge)  by  any  achieued :  And  alfo  it  is  moft  hard ,  in  thefe  our  drefy  dayes, 
to  fuch  rare  and  ftraunge  Artes, to  wyn  due  and  common  credit :  N euertheles ,  if, 
for  my  fincere  endeuour  to  fatiffie  your  honeft  expectation ,  you  will  but  lend  me 
your  thakefull  mynde  a  while:and,to  fuch  matter  as,for  this  time, my  penne  (with 
fpede)is  hable  to  deliuer,  apply  your  eye  or  eare  attentifely  :  perchaunce ,  at  once, 
and  for  the  firft  falutyng,this  Preface  you  will  finde  a  leffon  long  enough.  And  ei¬ 
ther  you  will, for  a  fecond  ( by  this  )  be  made  much  the  apter:  or  fhortly  become, 
well  hable  your  feiucs,  of  the  lyons  claw ,  to  coniedtire  his  royall  fymmetrie ,  and 
farderpropertie  .  Now  then, gentle, my  frendes,  and  coufitrey  men,Turne  your 
eyes, and  bend  your  myndes  to  that  dbdrine ,  which  for  our  prefent  purpofe ,  my 
fimple  talent  is  hable  to  veld  you. 

^11  thinges  which  are,&  haue  beyng,  are  found  vnder  a  triple  diuerfi tie  generall. 

For, either, they  are  demed  Supematurall,Naturall,or,of  a  third  being.Thinges 
Supematurall,  are  immatcriall,  fimple,  indiuifible, incorruptible,  &  vn  changeable. 
Things  Naturall,  are  materiah,compounded,diuifiblc, corruptible,  and  chaungea- 
ble.Thinges  Supernatural], are, of  the  minde  onely,cOmprehended:Things  Natu¬ 
rall, of  the  fenfe  exterior, ar  hable  to  be  perceiued.In  thinges  Naturall, probability 
and  coniedure  hath  place: But  in  things  Supematurall, chief  demoftration,&  mof| 
fure  Science  is  to  be  had.  By  which  properties  &  comparafons  of  thefe  two,  more 
eafilv  may  be  defcribed,the  ftate, condition,  nature  and  property  of  thofe  thinges, 
which, we  before  termed  of  a  third  being:  which, by  a  peculier  name  alfo,are  called 
Thjnges  at  hematic  ail.  F  or,th  efc,bey  ng  (in  a  maner)middle,  betwene  thinges  fu- 

pernaturall  and  naturalharc  not  fo  abfolute  and  excellent, as  thinges  fupernaturah 
Nor  yet  fo  bafe  and  groffe,as  things  naturall: But  are  thinges  immateriall :  and  ne- 
uertheleffc,by  materiall  things  hable  fomewhat  to  be  fignified  .  And  though  their 
particular  Images  ,  by  Art,areaggt  egableand  diuilible :  yet  the  generall  Formes^ 
nouvithftandyng,areconftant,vnchaungeable,vntrafformable,andincorruptible. 
Neither  of  the  fenfe, can  they, at  any  tyme, be  perceiued  or  iudged.Nor  yet, for  all 
that,  in  the  royall  mynde  of  man, firft  conceiued.But,furmountyng  the  imperfedio 
o f c on i edur  e,  weeny  n  g and  opinion:and commyng fhort  ofhigh  teteUcdaall  c5- 
cepti6,are  the  Mercurial  fruite  of  Dianceticall  difcourfe,in  perfect  imagination  fub- 
fiftyng.  Ameruaylousnewtralitiehaue  thefe  thinges  ^Mathematical! .  and  alfo  a 
ftraunge  participate  betwene  thinges  fupematurall,immortall,intelle(5lual,  fimple 
and  indiuifible:and  thynges  naturall, mortall/enfible, compounded  and  diuifible. 
Probabilitie  and  fenfible  profe,may  well  feme  in  thinges  naturall: and  is  commen¬ 
dable:  In  Mathematical!  reafoninges,a  probable  Argument,  is  nothyng  regarded: 
nor  yet  the  teftimony  of  fenfe, any  whit  credited :  But  onely  a  perfed  demonftra- 
tion,  oftruthes  cemine,neceftary,and  inuindbkoWniuerfally  and  neceffaryly  con¬ 
cluded  2 

i  .  ’  •  ,  .  •»  '  031 


John  Dee  his  MatKematicall  Prasfkce* 


eluded:  is  allowed  as  fufficientfor  an  Argumentexa&Iy  and  purely  Mathematical. 

Of  Mathematical!  thingesgre  two  prin  cipall  kindcs :  namely,  Numbagmd Mag- 
nitnde.Tiumberfve  define, to  be, a  eertayne  Mathematical!  Sumc}of  Fnits.  And, an 
Jjfriity  is  that  thing  Mathematical!,  Indiuifible ,  by.  participation  of  fome  likenes  of 
whofe  property , any  thing,which  is  in  deede,oris  counted  O  ne^may  refonabiy  be 
called  One .  We  account  an  Fnit^z.  thing  Mathematicall ,  though  it  be  no  Number, 
and  alfo  indiuifible.'bccaufe^fi^materially^Number  doth  confifi  V  which  9  princi¬ 
pally  ,  is  a  thing  Mathematical l.  Magnitude  is  a  thing  Mathematicall  y  by  participation 
of  fome  likenes  of  whofe  nature  ,  any  thing  is  iudged  long ,  broade,  or  thicke .  A 
thicke  Magnitude  we  call  a  Bolide. ,  m  a  Body . WhazMagmtudefo  euer,i$  Sblide  or. 
Thicke,is  alfo  broade,&  long.  A  broade  magnitude, we  call  a  Superficies  or  a  Plaine, 
Huery  playne  magnitudc,hath  alfo  length.  A  long  magnitude,  wetermea  Line.  A 
Line  is  neither  thicke  nor  broade,  butonely  long :  Euery  ceitayne  Line,  hath  two 
endes;The  endes  of  a  line, are  Pointes  called.  A  Point  f  s  a  thing  Mathematicall ,  indb 
uilible, which  may  haue  a  certavne  determined  fituation .  If  a  Povnt  mouefrom  a 
determined  fituation ,  the  way  wherein  it  moued,  is  alfo  a  Line :  mathematically 
produced,  whereupon, of  th e  auncicn t  Mathematicicn s,a  Line  is  called  the  race  or 
courfe  of  a  Point .  A  Poynt  we  define ,  by  the  name  of  a  thing  Mathematical!} 
though  it  be  no  Magnitude,  and  indiuifible:  becaufeitis  thepropre  ende,  and 
bound  of  a  Line :  which  is  a  true  Magnitude .  And  Magnitude we  may  define  to  be 
that  thing  Mathematical f  which  is  diuifible  for  eucr,in  partes  diuifible, long, broade 
or  thicke .  Therefore  though  a  Poynt  be  no  ^Magnitude,  yetT erminatiuely  weree- 
ken  it  a  thing  Mathematicall  (as  I  layd)by  reafon  it  is  properly  the  end  </  and  bound, 
of  a  line.  . 


*> 

Number. 

Note  the  rt)oricf 
Unit, to  ext  refit 
the  Grel>e  Mo-~ 
not  Vni- 
tie  :  at  tee  kau* 
d/l,  commonly, 
till  nowjvfed. 

Magnitude. 

»> 


A  point, 
oi  Line . 


N either  ‘Number, not  OMagnUtudeJiam  any  Materialitie.  Firft,we  will  confidef 
of  Number,and  of  the  Science  Mathematicall ,  to  it  appropriate, called  Arithmetifoi 
and  afterward  of  Magnitude fmdhis  Science,  cdkdGeometrie.  But  that  name  con* 
tenteth  me  not:  whereofa  wordor  two  hereaftcr  fhall  be  fayd .  How  Immaterial! 
and  free  from  all  matter  , Number  is  y  who  doth  hot  perceaueW  yea,  who  dotbndl 
wonderfully  woder  at  ME or,  neither  pure  Element,  nor  Ariftoteles, Quinta  Ejfcntia, 
jshable  to  feme  for  Number, ashis  propre  matter.  Nor  yet  the  puritie  and  fimple- 
nes  of  Subfiance  Spiritual!  or  Angelicall  ,  will  be  found  propre  enough’  thereto. 
And  therefore  the  great  &  godly  FhilpCofk&i  Anitms  Boetws,  fayd :  Omnia  qu&cimfp 
apriffidua  rerum  natutd  cmfiru£htfmty  Numemrumvidentur  ration? format a.  Hoc  enim 
ftpit principal?  iri  amnio  €  on  dh  oris  Exemplar .  THatfS  ^  thinges  ( 1 t>hich  front 

the  Titery  firft  qrtgtnall  being  of  ihiiigef0 -Mite  bene  framed  and  made) 
do  appear?  to  be  Formed  by  the  reajon  of  jSlumbers  V  For  this  'toas  the 
princi pall  example  or  patterne  in  the  mind?  of  the  Creator  .  O  comfor¬ 
table  allurcmcnr,  O  rauilhing  perlwafion,  to  deale  with  a  Science,  whofe  Subie^ 
is  fo  Auncientdo  pure,fo  exyelienWfc  lurmouii  ting  all  creature5,fb  yfed  of  the  Al¬ 
mighty  and  incomprehenfiblc  wifdomc  of  the  Creator5  in  the  diftimfi  creation  of 
allcreatures: in  all  their  diftinbt  partes,  propertiesAnatures ,  and  vertues,  by  order, 
and  moft  abf0lute  number,brought,from  Npthingyto  the  Formalities their  being 
andfiate.By  Numbos  propertic  therefore, ofvs,by  all  pofiible  meanes,(to  the  per- 
fe#ion  of  the  Science  )  learned, we  may both  winde  and  draw  our  felues  into  th^ 
inward  and  deepe  fearch  and  ve>v0of  all  creatures  diftimfi  vertues, natures,  proper- 
tie^, and  Formes:  And  alfojfarderjarifejcfimejalcend^nd  mount  vp  (  with  Specula. 
tiue  winges  }  in  fpirit,  to  behpld  in  the  Gks  of  Creation,  the  Forme  of  femes  y  the 
Exemplar  Number  ofall  thing!esiiV^^4^/bpdl  vifible  and  inuifible  .•  mortall  and 
,  7  -  -  *♦]..  immortal! 


lohnD  ee  his  Matliematicall  Preface . 

>  immortall,C  orporall  and  SpirituallJPart  ofthis  profound  and  diuine  Science,  had 
Joachim  the  Rrophefier  attey  ned  vnto :  by  ‘Numbers  Formal!., Natural!,,  and  Rational!, 
forfeyng,concludyng,and forfheWyng  great  particular  euents ,  long  before  their 
comming.His  bookes  yet  remainyng, hereof, are  good  profe:  And  the  noble  Earle 
of  Miranduk,(\>&d.zs  thar,)a  fufficient  witnefle:that  loachim,in  his  prophef.es,  proce* 
trim*  1488*  4ed by  no  other  my, then  by  Numbers  Formal!.  And  this  Earle  hym  felfedn  Rome,*fet 

vp -poo. Conduftons,in  all  kinde  of  Sciences, openly  to  be  difputed  of  :and  among 
the  reft,  in  his  Conclufions  CMathematicall,  (in  the  eleuenth  Conclufton  )  hath  in 
Latin,this  Englifh  fenten cc. By  Numbers, a  my  is  had  ,  to  thcfearchyng  out, dud  vnder - 
jlandyng  of euery  thyng ,  hable  to  he  knowen .  For  the 'verifying  of  which  Conclufon ,  I  pro - 
mife  toaurfvere  to  the/  4.  glujtflions^mder  written, by  the  way  of  Numbers  .Which  Co- 
clufions,I  omit  here  to  rehearle:  afwell  auoidyng  fuperfluousprolixitie:as  ,  by- 
caufe  Joannes  Ficus, workes ,  are  commonly  had.  Bht,in  any  cafe, I  would  wifh  that 
thofe  Conclufions  were  red  diligently  ,  and  perceiued  of  fuch,as  are  earn  eft  Ob- 
feruers  and  Confiderers  of  the  conftant  law  of  nubers:  which  is  planted  in  thyngs 
Naturall  andSupernaturalljandis  preferibed  to  all  Creatures, inuiolably  to  be 
kept*For,fo,beftdes  many  other  thinges  ,  in  thofe  Conclufions  to  be  ma'rked,it 
would  apeare,how. fincerely,&  withinmy  boundes,I  difclofe  the  wonderfull my- 
fteries,by  numbers,to  be  atteynedvnto.  ■ 

Of  my  former  wordes,eafyit  is  to  be  gathered,that  Number  hath  a  treble  ftatei 
One, in  the  Creatorum  other  in  euery  Creature(in  refped  of  his  complete  conftb 
tution;)aud  the  third,in  Spiritualland  Angelicall  Myndes,and  in  the  Soule  of  ma. 
In  the  firft  and  third  ^ant,Numher  >  is  termed  N timber  Numbryng.  But  in  all  C  rea- 
tures,otherwife,7V//w^,is  termed  NuberNumbred.  And  in  our  Soule, N  uberbea- 
teth  fuch  a  fwaye^and  hath  fucit  anaftmitie  therwith  .■  that  fome  of  the  old  Philofo - 
pbers  t&npftt,Makr$otde,to  brgaNumbefmouyng  kfitfe:  And  in  dede,in  vs,  though  it 
beu  Yery  Accident:  yet  fuch  aii^ Accident  it is;thait  before  all  Creatures  it  had  per-; 
£b<S?  beyng,  in  thc.C  vcatorfcmpkdma&y.NumberNumbryng  therfore,is  the  difere- 
tion  difccrning,and.  diftincHng  ofthinges*  But  in  God  the  Creator ,  This  dilcre- 
tion,in  thebeginnyng,produ€ed\orderly  anddiftimftly  allthinges.  Eor  his  Num- 
foy^,then,was  his  Creatyngbf  all.thinges.  Aral  his  QontmnaWi Numbryng ,  of  all 
thinges,is  the  Gonfemation  of  them  in  being:  And^where  and  when  he  will  lacke 
an :Fnit:  there  and  thaivthat  particular  thyng ^ftialbejDi/c^fez/ddefe  I  ftay.But  our 
Seumllyng^difti,n'#yng^nd^»i^r)fj^,createth'ii6thyngrbutofMultttudexom 
fidered,raakedi  Gertaine and  diftinCtdetermination  ,  And  albeit  thefe  thynges be 
Whty  and .truces. ofgreat , ,yer  (.b^tbe  infinite  gso^esoftheAl- 
mighty  Zemarie,)  Artmqall  Methods  and  eafyw^yes  are  made ,  by  which  the  ze- 
Philofopher jiifay  wyn  nere  this  fliuerifti  iddltMs  Moiintay  rie  of  Cohtempk- 
tionrand  momthen^Rnitempfatibh.Andallbkhotigh  Number, bed.  thyng  To  I  in- 

iV»  T  i  1 1  f  &  A  frvv-l  4 1 1*  t  rnf'.  Ka  r  rl  IT  v  tT  Vi  m  1 1  A  ft 


KF 


lower, to  thynges  fenftbly  pefeeiuedras  of  a  mbmentiinye  found  e  iterated:  then  td 
the  lcaft  thynges  that  may  bc'fCen,niimerable:Arid  at  length, (moff  groffely,)  to  a 
multitude  of  any  corporall  thynges  feeh,or  felt:  and  fb,of  thefe  grofte  and  fenftble 
thynges, we  are  trayhed  to  leame  a  ce'rtaine  Image  or  likenes  of  numbers :  and  to 
vfe  Arte  in  them  to  our  pleafufe  arid  proffit.So  grofte  is  our  conuerfation,  and  dull 
iSourapprehenfion:  while  mortal!  Senfe,  in  vs,  ruleth  the  common  wealth  of  our 
litle  world.Hereby  we  fay, Three  Lybns,afe  three:  or  a  F ernarie .  Three  Egles^are 
three, or  a  F ernarie  •  Which*  T ernaries ,  are  eche,the  Fnion,  knot, and  Zniformitie,o? 
three  diferete  and  diftimft  Vnit-i* That^ is, we  may  ineche  T ernarie ,  thrife ,  feuerafty 
pointe,and  fhew  a  pan, One, One, and  One.  Where,  in  Numbryng, we  fay  One, two. 


Three. 


John  Dee  his  Mathematical!  Preface. 

Three .  But  how  farre, thefe  vifible  Ones ,  do  differre  from  our  Indiuifible  Vhits 
(in  pure  Arithmetike, principally  confidered)no  man  is  ignorant  .  Yet  from  thefe 
groife  and  materiall  thynges,may  we  be  led  vp ward, by  degrees, fofinformyng  our 
rude  Imagination,toward  the  coceiuyng  of  Numbers/dbfoluttly ( :N  ot  fuppoling, 
nor  admixtyng  any  thyng  created,Corporall  or  Spiritual!, to  fupport,conteyne,or 
reprefent  thole  Numbers  imagined : )  that  at  length,  we  may  be  hable ,  to  finde  the 
number  of  our  ownc  name ,  glorioully  exemplified  and  regiftred  in  the  booke  of 
the  Trinitk  moft  blelfed  and  asternall. 

But  farder  vnderftand,that  vulgar  PraCtifers,haue  N umbers  otherwife,  in  fun- 
dry  Confiderations:and  extend  their  name  farder, then  to  N umbers ,  whole  leaft 
partis  an  Vnit .F  or  the  common  Logili,R cckenmaftcr,  or  Arithmeticien,  in  hys  v- 
fing  of  N  umbers :  of  an  V  nit,imagineth  lelfe  partes :  an  dcalleth  them  Fractions.  As 
of  an  Vnit ,  he  maketh  an  halfe,and  thus  noteth  it, A.,  and  fo  of  other,  (infinitely  di¬ 
uerfe)  partes  of  an  Vnit.  Yea  and  farder, hath,  Fractions  of  Fractions.  &c .  And,foraf 
much, as,  Addition  >  Subfir  action ,  Multiplication ,  Dim  [ion  and  Extraction  of  Rotes,  are 
the  chief, and  fufficient  partes  of  Ar Ahmet  ike  .*  which  is ,  the  Science  that  demon  fir  a* 
teth  the  properties, of Numbers, and  all  ope  ratios ,  in  numbers  to  be  performed /How  often, 
therfore, thefe  fiue  fundry  fortes  of  Operations,  do ,  for  the  moll  part, of  their  cxc-  „  Note, 
cution,dilfcrre  from  the  fiue  operations  oflike  generall  property  and  namefin  our  ,, 

Whole  numbers  praCtilable,So  often  ,  (for  a  more  difiinCt  doctrine  )  we,  vulgarly  „ 
accountand  name  it, an  other  kynde  of  Anthmetike  .And  by  this  realon.-the  Com 
fideration,dodtrine,and  working, in  whole  numbers  onely:  where,  ofan  Vnit, is  no 
Idle  part  to  be  allowed:  is  named  (as  it  were)an  Arithmetike  by  it  lelfe .  And  fo  of 
the  Arithmetike  of  Fractions. In  lyke  forte, the  neceflary,wonderfull  and  Secret  doc¬ 
trine  of  Proportion ,  and  proportionalytie  hath  purchafed  vnto  it  lelfe  a  peculier  2, 
maner  of  handlyng  and  workyng:and  fo  may  feme  an  other  forme  of  Arithmetike. 
Moreouer,the  Astronomers  for  fpede  and  more  commodious  calculation,haue  de-  2 . 
uifed  a  peculier  maner  of  orderyng  n fibers, about  theyr  circular  motions,  by  Sexa-  3 
genes, and  Sexagefmes.By  Signes, Degrees  and  Minutes  &c .  which  commonly  is 
called  the  Arithmetike  of  Agronomical  or  Phificall  Fr actions.  That,  haue  I  briefly  no¬ 
tedly  the  name  of  Arithmetike  Circular.  Bycaufe  it  is  alfo  vfed  in  circles,nc  t  Afiro- 
wmicall. c. Pradtile  hath  led  Numbers  farder ,  and  hath  framed  them, to  take  vpon  A. . 
them  ,  the  fhew  of  Magnitudes  propertie:  Which  is  Incommenfurabilitie  and  Irratio - 
nalitie.  (For  in  pure  Arithmetike, cm  Vnit, is  the  common  Meafurc  of  all  N  umbers.) 

A nd,here,N fibers  are  become, as  Lynes,Playnes  and  Solides:  fome  tymes  Ratio- 
nail, fome  tymes  Irrationall.  And  haue  propre  and  peculier  characters, (as  c£. 

and  fo  of  other.  Which  is  to  lignifie  Rote  Square ,  Rote  Cubik:and fo  forth:  )& propre 
and  peculier  falhions  in  the  fiue  principall partes:  Wherfore  the  pradifer,e!temeth 
this,a  diuerfe  Arithmetike  from  the  other .  PraCtife  bryngeth  in,here,diuerfe  com- 
poundyngofNumbers:  asfome  tyme, two, three, foure(or more) Radicall  nubers5 
aiuerlly  knit, by  lignes,  ofMore  &  Lelfe:as  thus  12  +  v'ct  i5.0r  thus  ig 
-b  v/cc'i^—bb‘2.  &c.  And  fome  tyme  with  whole  numbers,  orfraCtions  of  whole 
Number,amog  them :  as  20  33-V^  xo.  —  -j-CcCp. 

And  fo,  infinitely ,  may  hap  the  varietie.  After  this  :  Both  the  one  and4 the  other 
hath  fractions  incident:andfo  is  this  Arithmetike  greately  enlarged, by  diuerfe  ex- 
hibityng and  vfe  of  Compofitions  and.mixtyriges .  Confidcr how,  I (beyng  deli- 
rous  to  deliuer  the  ftudentfrom  error  and  Cauillation)do  giue  to  this  PraCtife, the 
name  of  the  Anthmetike  of  Radicall  numbers:  Not, of  Irrationall  or  Surd  Numbers; 
which  otherwhile,  are  Rationall :  though  they  haue  the  Signeofa  Rote  before 
b  *.ij*  them., 


lohn  Dee  his  Mathematical!  Preface. 

them,  which,  Arithmetike  of  whole  Numbers  moft  vfuall ,  would  lay  they  had  no 
fuch  Roote:  and  fo  account  them  Surd  Numbers:  which,  generally  (poke,  is  vntrue; 
as  Euclides  tenth  booke  may  teach  you.  Therfore  to  call  them ,  generally ,  Radical! 
Numbers,  (by  reafon  of  the  ligne  / .prefixed,)  is  a  fure  way :  and  a  diffident  generall 
diftinftion  from  ail  other  ordryng  and  vfing  of  N umbers :  And  yet  (  befide  all 
this)Confider :  the  infinite  delire  of  knowledge ,  and  incredible  power  of  mans 
Search  and  Capacitye:  how,  they,  ioyntly  haue  waded  farder  (  by  mixtyng  offpe- 
culation  and  praftife)and  haue  found  out ,  and  atteyned  to  the  very  chief  perfec¬ 
tion  (almoft)  of Numbers  Pradicali  vfe. Which  thing,is  well  to  be  perceiued  in  that 
great  Arithmetical!  Arte  of  Aquation :  commonly  called  the  Rule  of Cojf.  or  ^Alge¬ 
bra.  The  Latines  termed  it,Regulam  Ret  &  Cenfus ,  that  is ,  the  rftile  of the  thyng 
and  his  lvalue*  With  an  apt  name :  comprehendy  ng  the  firft  and  laft  pointes  of  the 
worke .  And  the  vulgar  names  ,  both  in  Italian ,  Frenche  and  Spanilh,depend(in 

namyng  it,)  vpon  the  lignification  of  the  Latin  word,i?«:  A  f/;/wg:vnleaft  they  vfe 
the  name  of  ^Algebra.  And  therm (commonly)is  a  dubble  crror.The  one,of  them, 
which  thinke  it  to  be  of  Geber  his  inuentyng:  the  other  of  fuchascallit  Algebra. 
For, firft, though  Geber  for  his  great  fkill  in  N umbers, Geometry,  Aftronomy ,  and 
other  maruailous  Artes,moughthauefemed  hable  to  haue  firft  deuifedthefayd 
Rule:  and  alfo  the  name  carryeth  with  it  a  very  nere  likenes  of  Geber  his  name :  yet 
true  it  is, that  a  Greke  Philofopher  and  Mathematicien,named  Diophantus,  before 
Geber  his  tyme,wrote  i3.bookes  therof  (  of  which ,  fix  arc  yet  extant :  and  I  had 
*Anno,is^t3,  them  to  *  vfe,of  the  famous  Mathematicien,and  my  great  frende,  Petrus  <j\€ont. au¬ 
reus  : )  And  fecondly,the  very  name, is  Algiebar, and  not  Algebra :  as  by  the  Arabien 
Autcen, may  be  proued:  who  hath  thele  precife  wordes  in  Latine,by  Andreas  Alpa~ 
^tffmoft  perfeft  in  the  Arabik  tung  )  fo  tranflated .  Scientia faciendi  Algiebar  & 
Almacbabel.  i.  Scientia  inueniendi  numerum  ignotum,ptr  additionem  Numeri ,  &  diuifeo . 
nem  &  aquationem. Which  is  to  fay:T he  Science  of^oorhyng  Algiebar  and  yfl* 
machabelfhdzisjhc Science  offindyng an  Imknowen  number  ,  by  Addyng  of a 
Number,  er  Diuijton  aquationMcrc  haue  you  the  name :  and  alfo  the  prin- 
cipall  partes  of  th  e  R  u!e,touched.T  o  name  it0The  rule, or  Art  of  Aquation, doth  fig- 
nifie  the  middle  part  and  the  State  of  the  Rule .  This  Rule,  hath  his  peculier  Cha- 
rafters: and  the  principal  partes  of  Arithmetike, to  it  appertayning,do  differre  from 
the  other  Arithmetical!  operations. This  Arithmetike,  hath  Nubers  Simple, Copound, 
Mixtrand  Fraftions, accordingly.  This  Rule,  and  Arithmetike  of  Algiebar,  is  fo  pro¬ 
found,  fo  generall  and  fo  (in  maner)  conteynedi  thewhole  power  of  Numbers 
Application  prafti call:  that  mans  witt, can  deale  with  nothyng,more  proffitable  a- 
bout  numbers :  nor  match ,  with  a  thyng ,  more  mete  for  the  diuine  force  of  the 
Soule, (in  humane  Studies, affaires, or  exercifesjto  be  tryedin.  Perchaunceyou 
looked  for, (long  ere  now,)  to  haue  had  fome  particular  profe,  or  euident  teftimo- 
ny  of  the  vfe, profit  t  and  C  ommodity  of  Arithmetike  vulgar,  in  the  C  oiftmon  lyfe 
and  trade  of  men.Therto,then,I  will  now  frame  my  felfe :  But  herein  great  care  I 
haue,  leaft  length  of fundryprofes,  might  make  you  deme,  that  eitherl  did  mi£ 
doute  your  zelous  mynde  to  vertues  fcliole :  or  els  miftruftyour  hable  witts ,  by 
fome,to  geffe  much  more.  A  profe  then,foure,fiue,or  fix,  fuch ,  will  I  bryng ,  as 
any  reafonable  man,thenvith  may  be  perfuaded,to  loue  &  honor ,  yea  karne  and 
exercife  the  excellent  Science  of  Arithmetike. 

And  firft:  who, nerer  at  hand,  can  be  a  better  witnefte  of  the  frute  receiued  by 
Arithmetike,  Aon  all  kynde  ofMarchants  i  Though  not  all, alike,  either  nede  it,or 
vfe  it.How  could  they  forbeare  the  vfe  and  helpe  of  the  Rule ,  called  the  Golden 

Rule? 


Iohn  Dee  his  Mathematical!  Preface. 

Rulec’Simple  and  Compounde:both  forward  and  backward  <  How  might  they 
miffe  Arithmetkall helpe  in  the  Rules  ofFelowfhyp:  either  without  tyme,  or  with 
tyme'and  betwene  the  Marchant&  his  Fa&or  i  The  Rules  ofBartering  in  wares 
onely:  or  part  in  wares,  and  part  in  money,  would  they  gladly  want  i  Our  Mar- 
chant  venturers,  and  Trauaylersouer  Sea  ,  how  could  they  order  their  doynges 
iuftly  and  without  Ioffe ,  vnlcaft  ccrtainc  and  generall  Rules  for  Exchauge  of  mo¬ 
ney  5and  Rechaunge, Were, for  their  vfe,deuifed  f  TheRuleofAlligation,in  how 
fundry  cafes,doth  it  conclude  for  them,fuch  precife  verities, as  neither  by  natural! 
witt ,  nor  other  experience, they, were  hable,  els, to  know  <  And ( with  the  Mar- 
chant  then  to  make  an  end  )  how  ample  &  wonderfull  is  the  R  ule  of  Falfe  pofiti- 
ons  <  efpecially  as  it  is  now,  by  two  excellent  Mathematiciens  (  of  my  familier  ac- 
quayntance  in  their  life  time ) enlarged  <  I  meane  Gemma  Frifius, and  Simon  Jacob . 
Who  can  either  in  brief  conclude ,  the  generall  and  Capitall  R  ulest  or  who  can  I- 
magine  the  Mvriades  of  fundry  Cafes,and  particular  examples,in  Ad  and  earn  eft, 
continually  wrought, tried  and  concluded  by  the  forenamed  Rules, onely  <  How 
fundry  other  Arithmeticall  praclifes ,  are  commonly  in  Marchantes  handes,and 
knowledge: They  them  felues,can,at large, teftifie. 

The  Mintmafter,and  Goldfinith,in  their  Mixture  of  Metals ,  either  of  diuerfe 
kindes,qr  diuerfe  values.-how  are  they, or  may  they,exadly  be  directed ,  and  mer- 
uailoufly  pleafured,if  Arithmetike  be  their  guided  And  the  honorable  Phificias, 
will  gladly  confeffe  them  felues much  beholding  to  the  Science  of  Arithmetike^ 
and  that  fundry  way es :  But  chiefly  in  their  Art  of  Graduation ,  and  compounde 
Medicines,  And  though  Galenas,  Auerrou,Armldiis ,  Lidias ,  and  other  haue  pu¬ 
blished  their  pofitions ,  afwell  in  the  quantities  of  the  Degrees  aboue  Tempera* 
ment ,  as  in  the  Rules ,  concluding  the  new  Forme  refulting :  yet  a  more  precife, 
commodious, and  eafy  (JMethodf*  extantrby  a  Countreyman  of  ours  ('aboue  200. 
yeares  ago)inuented.  And  forafmuch  as  I  am  vncertaine ,  who  hath  the  fame: 
or  when  that  litle  Latin  treatife,  (as  the  Author  writ  it, )  fhall  come  to  be  Printed: 
(Both  to  declare  the  defire  I  haue  to  pleafure  my  Countrey,wherin  I  may  :  and  al- 
fo,for  very  good  profe  of  Numbers  vfe,in  this  moftfubtile  and  frutefull ,  Philofo- 
phicall  Conclufion, )  I  entend  in  the  meane  while ,  moft  briefly,and  with  my  far- 
der  helpe, to  communicate  the  pith  therof  vnto  you. 

Firft  defcribe  a  circle :  whole  diameter  let  be  an  inch .  Diuide  the  Circumfe¬ 
rence  into  foure  equall  partes.  Fro  the  Center,  by  thofe  4.fedions,extend  fright 
lines  :  eche  of  ^.inches  and  a  halfe  long :  or  of  as  many  as  you  Iifte,aboue  4.  with¬ 
out  the  circumference  of  the  circle :  S  o  that  they  fhall  be  of  4-inches  long  (  at  the 
leaft)  without  the  Circle .  Make  good  euident  markes,at  euery  inches  end.  If  yott 
lift,  you  may  fubdiuide  the  inches  againe  into  10.  or  i2.fmallerpartes,equall.  At 
the  endesofthe  lines,  write  the  names  of  the  4.  principall  elementall  Qualities. 
Hote  and  Colde ,  one  againft  the  other .  And  likewife  CMoyst  and  Dry,  one  againft 
the  other.  And  in  the  Circle  write  Temperate.  Which  T emperature  hath  a  good  La¬ 
titude  :  as  appeareth  by  the  Complexion  of  man .  And  therefore  we  haue  allow¬ 
ed  vnto  it,  the  forefay  d  Circle :  and  not  a  point  Mathematicali  or  Phyficalb 

Now,  when  you  haue  two  thinges  Mifcible  ,  whole  degrees  are  *  truely 
knowen  :  Ofneceflitie,  either  they  are  of  one  Quantitie  and  waight,  or  of  diuerfe. 
If  they  be  of  one  Quantitie  and  waight:  whether  their  formes,be  Contrary  Qua- 
lities,  or  of  one  kinde  (but  of  diuerfe  intentions  and  degrees)  or  a  T emperate,  and  a 
Contrary ,  T be forme  resulting  of  their  Mixture, is  in  the  Middle  hetwene  the  degrees  of 

the 


R» 


*Takefom 
part  of  LuUm 
counptylein 
his  booke  de 
QJEjJentk. 


lohn  Dec  his  Mathematical!  Preface. 

the  formes  mixt .  As  for  example, let  o*f,be  Moitt  in  the  firft  degree :  and  B ,  Dry 
in  the  third  degree .  Adde  i.  and  3.  that  maketh  4 :  the  halfe  or  middle  of  4.1s  2. 
*No;e,  T his  2.1s  the  middle,  equally  diftant  from  A  and  B  (  for  the  *  T emper ament  is  coun~ 
ted  none  .  And  for  it,  you  mud  put  a  Ciphre,  if  at  any  time,  it  be  in  mixture). 


HOTE 


Counting  then  from  B,  2.  degrees ,  toward  you  finde  it  to  be  Dry  In  the  firft 
degree :  So  is  the  Form  refulting  of  the  Mixture  of  ^,and  B ,  in  our  example.  I  will 
sreue  you  an  other  example .  Suppofe,  you  haue  two  thinges,as  C,and  D  and  of 
C,  the  Heate  to  be  in  the  4.degree :  and  of  D,  the  Colde,  to  be  remifle,euen  vnto 
the  Temperament .  N  ow,for  C,you  take  4:  and  for  D,you  take  a  Ciphre  h  which, 
added  vnto  4, yeldeth  one!y4.The  middle, or  halfe,  whereof,  is  2.  Wherefore  the 
Forme  refilling  of  C ,  and  D,  is  Hote  in  the  fecond  degree:  for,  2.  degrees,accoun- 
ted  from  C3  toward  D ,  ende  iufte  in  the  2.  degree  of  heate  .  Of  the  third  ma- 
nei‘,1  will  geue  alio  an  example:  which  let  be  this :  I  haue  a  liquid  Medicine  whole 
Nate.  Qualitie  ofheate  is  in  the  4-degree  exalted :  as  was  C,  in  the  example  foregoing: 
and  an  other  liquid  Medicine  I  haue  .•  whofe  Qualitie,  is  heate,  in  the  firft  degree. 
O  f  eche  o  f thefe,  I  mixt  a  like  quantitie  ,*  S  ubtrad  here, the  ldfe  fro  the  more  .•  and 
the  refidue  diuide  into  two  equall  partes  ••  whereof,  the  one  part,  either  added  to 
the  idle, or  fubtrafled  from  the  higher  degree,  doth  produce  the  degree  of  the 

Forme 


I  qIui  Dee  his  M  atliem  aticall  P  rceface. 

Forme  refulting, by  this  mixture  of  C,and  E .  As,iffrom  4.  ye  abate  1. there  rcfteth 
3.thehaifeof3-is  iff  ;  Addeto i.thisiQ-  ,*  youhaue2-L.  »  Orfubtradfromq. ' 
this  1  —  .*  you  haue  like  wife  2--  remayning .  Which  declarefh  5  the  Formereful- 

ting,  to  be  He  ate,  in  the  middle  of the  third  degree. 

But  if  the  Quantities  of  two  thingesCommixt,  be  diuerfe,  and  the  Intend-  M  TheSe- 
ons  (  of  their  Formes  Mifcible  )  be  in  diuerfe  degrees ,  and  heigthes.  (  Whether  C0„J 
thofe  Formes  be  of  one  kinde,or  of  Contrary  kindes,  or  of  a  Temperate  and  a  jtuic 
Contrary ,  What  proportion  is  of  the  lejfe  quant itie  to  the  greater,  the  fame  fall  be  of  the  ?> 
difference  jvhich  is  betwene  the  degree  of  the  F  orme  refulting ,  and  the  degree  of  the  greater  }  f 
quant  itie  ofthethingmifcible,  to  the  difference,  which  is  betwene  the  fame  degree  of  the  }j 
Forme  refulting^and the  degree  of  the  lefe  quantitie .  As  for  example .  Let  two  pound 
of  Liquor  be  geuen,  hote  in  the4.degree:&  one  pound  of  Liquor  be  geued,  bote  „ 
in  the  third  degree .  I  would  gladly  know  the  Forme  refulting,in  the  Mixture  of  }f 
thefe  two  Liquors.  Set  downe  your  nubers  in  order ,  thus. 

Now  by  the  rule  of  Algiebar,  haue  I  deuifed  averyeafie, 
briefe,  and  generall  maner  of  working  in  this  cafe  .  Let  vs 
firft,  fuppofe  that  Middle  Forme  refulting ,  to  be  iff  :  as  that 
Rule  teacheth .  Andbecaufe  (by  our  Rule,  here  geuen)  as 
the  waight  of  i .is  to  2  :  So  is  the  difference  betwene  4.  (the 
degree  of  the  greater  quantitie  )  and  12^  •  to  the  difference  betwene  12^  and  3.“ 

(the  degree  of  the  thing,  in  lefle  quatitie.  And  with  all,  12^,  being  alwayes  in  a  cer- 
taine  middell,  betwene  the  two  heigthes  or  degrees)  ,  For  the  firft  difference,  I  fet 
4 — tfdd  md  for  the  fecond,  I  let  iff — 3  .  And,  now  agaide,  I  fay,  as  i.is  to  2X0  is 
4— 1%6 to lXC~~3-  Wherforc,  ofthefe foure proportional! numbers, the firft and 
the  fourth  Multiplied, one  by  the  other,do  make  as  ‘much,  as  the  fecond  and  the 
third  Multiplied  the  one  by  the  other .  Let  thefe  Multiplications  be  made  accor¬ 
dingly .  And  of  the  firft  and  the  fourth, we  haue  iff — 3.andofthe  fecond  &thc 
third,  8 — 2%^.Whcrfore ,  our  Equation  is  betwene  iff — 3:  :  and  8 — 22^.  Which 
may  be  reduced,according  to  the  Arte  of  Algicbarras, here, adding  3 .to  eche  part, 
geueth  the  iEquation,thus,i2^—i  1—22^.  And  yet  againe,contra<fting,  or  Redm 
cing  it :  Adde  to  eche  part,  22^:  Then  haue  you  gff  squall  to  ii  :  thus  reprefen- 
ted  32^—11.  Wherefore, diuiding  n.by  3:  the  Quotient  is  3—  :  theCa/ovofour 

iff, Cof, or  T hing, firft  ftippofed.  And  that  is  the  heigth,  or  Intention  of  the  Forme 
rejtdting:  which  is,  Heate,  in  two  thirdes  of  the  fourth  degree :  And  here  I  fet  the 
Brew  of  the  workc  in  conclufion,  thus The  proufe  hereof  is  eafie-by  fubtrading 

3  .from  3-^-qefleth  . r 

ff  .Subtrade the 
fame  heigth  of  the 
Forme  refulting, 

(which  is  3—)  fro 
4;:  then  reijeth  W; 

•You  fee,  that 
cfS;  double  to  ff: 

as  2 .  f.  is  double  to  1.  &  So  ihoujd it  be by  the: rule  here  geuen .  H ote .  As  yoU  ad¬ 
ded  to  ech  e  part  of  the  Equation,  3 :  fo  if  ye  firft  added  to  eche  part  2%,  it  would 
hand,  32^— 3— .8 .  And  now.  adding  to  eche  part  3  .•  you  haue  (as  afore)  32^=11. 

And  though  I,  here,fpeake  dnely  of  two  thyngs  Mifcible:  and  moft  common- 
ly,mo  then  th  rec,fourc,fiue  or  iix,f  &c.)are  to  be  Mixed:  (and  in  one  Compound 

*.  iiijf.  to 


l  2. 

Hote.  4. 

f.  I. 

Hote. 

John  Dee  his  Mathematical!  Prseface. 

to  be  reduced.-&  the  Forme  refultyng  of  the  fame,  to  fettle  the  turne)yet  thefe  Ku~ 
Nvte,  les  are  fufficientrduely  repeated  and  iterated.In  procedyng  firft,  with  any  two  rand 
then,  with  the  Forme  Refulting,and  an  other ;&  fo  forth:For,  the  laft  workc,  con¬ 
clude  th  the  Forme  refultyng  of  them  all  .-I  nede  nothing  to  fpeake,  of  the  Mixture 
(here  fuppofed)  what  it  is.Common  Philofophie  hath  defined  it ,  faying,  uw/xtio 
tjl  mifcibilium ,  alter atorum ,  per  minima  coniunclorum,Vnio .  Euery  word  in  the  de-> 
finition,  is  of  great  importance.  I  nede  not  alfo  fpend  any  time,  to  fhew,how,the 
other  manner  of  diftributing  of  degrees, doth  agree  to  thefe  Rules.  N  either  nede  I 
of  the  farder  vfe  belonging  to  the  Croffe  of  Graduation  (before  defcribed)in  this 
place  declare, vnto  fuch  as  are  capable  of  that,which  I  haue  all  ready  fay  d.  N  either 
yet  with  examples  ipecifie  the  Manifold  varieties ,  by  the  forefayd  two  gene- 
rall  Rules,to  be  ordered.  The  witty  and  Studious, here, haue  fufficient:  And  they 
which  are  not  hable  to  atteinc  to  this, without  liuely  teaching ,  and  more  in  parti- ' 
cular:  would  haue  larger  difcourfing,then  is  mete  in  this  place  to  be  dealt  withalh 
And  other(perchaunce)with  a  proude  fhuffe  will  difdaine  this  litlerand  would  be 
vnthankefull  for  much  more .  I,therfore  conclude :  and  wifh  fuch  as  haue  modeft 
and  earneft  Philofophicall  mindes,to  laude  God  highly  for  thisrand  to  Meruayie, 
that  the  profoundeft  and  fubrileft  point,  concerning  Mixture  of  Formes  and  Quali¬ 
ties  Flat urall, is  fo  Matchtand  maryed  with  the  moflfimpIe,eafie,and  fbort  way  of 
the  noble  Rule  of  Algiebar.  Who  can  remaine ,  therfore  vnperfuaded,to  louc,a- 
low,and  honor  the  excellent  Science  of' Arithmetike  *  For, here, you  may  perceiue 
that  the  litle  finger  of  Arithmetike, is  of  more  might  and  contriuing,then  ahun- 
derd  thoufand  mens  wittes,of  the  middle  forte ,  are  hable  to  perfourme,  or  truely 
to  conclude,  with  out helpe  thereof. 

Now  will wefarder,by  the  wife  and  valiant  Capitaine,be  certified,  what  helpe 
he  hath, by  the  Rules  of Arithmetike /in  one  of  the  Artes  to  him  appertaining;  And 
Taxi  ix?.  „  of  the  Grckes  named  TcwlixiThat  is ,  the  Skill  of  Ordring  Souldiers  in  Battell  ray 
5,  after  the  beft  maner  to  all  purpofes.This  Art  fo  much  dependeth  vppon  N  umbers 
vfe,and  the  Mathematicals,  that  J&lianus  ( the  beft  writer  therof, )  in  his  worke,to 
the  Emperour  Hadrianus  ,  by  his  perfection,  in  the  Mathematicals,(beyng  greater, 
then  other  before  him  had,)  thinketh  hisbooke  topafle  all  other  the  excellent 
workes,written  of  that  Art,vnto  his  dayes.For,ofit,  had  written  ALneas :  Cyneas  of 
T hefdy :  Pyrrhus  Epirota:zx\A  Alexander  his  forme:  Clear ebus:  Paufanias :  Euangelus: 
Polybius  fdimlitt  frende  to  Scipio :  Eupolemus:  Iphicrates ,  Poffdonius:  and  very  many 
other  worthy  Capitaines ,  Philofophers  and  Princes  of  Immortall  fame  and  me¬ 
mory :  Whole  fayrefl  floure  of  their  garland  ( in  this  feat)  was  ^Arithmetike ;  and  a 
litle  perceiuerance,in  Geometricall  Figures .  But  in  many  other  cafes  doth  ^Arith- 
metike  (land  the  Capitaine  in  great  ftede.  As  in  proportionyng  ofvittayles ,  for 
the  Army, either  remaining  at  a  Ray  :  or  fuddenly  to  be  encrcafed  with  a  certaine 
number  ofSouldiers.-and  for  a  certain  tyme.Or  by  good  Art  to  diminifh  his  com- 
pany,to  make  the  victuals, longer  to  ferue  the  remanent,  &  for  a  certaine  determi- 
ned  tyme :  if  nede  fo  require.  And  fo  in  fundry'  his  other  accounted,  Recke- 
ninges,Meafurynges,and  proportionynges,the  wife, expert, and  CircumfpeCt  Ca¬ 
pitaine  will  affirme  the  Science  of  Arithmetike ,  to  be  one  of  his  chief  Counfaylors, 
direCtersand  aiders*  Which  thing(by  good  meanes)was  euident  to  the  Noble, 
the  Couragious ,  the  Ioyall ,  and  Curteous  John ,  late  Earle  ofWarwickc.  Who 
was  a  yong  G  entlernan ,  throughly  knowne  to  very  few .  Albeit  his  lufty  valiant- 
nes,force,and  Skill  in  Chiualrous  feates  and  exercifes:his  humblenes,andfrende- 
lynes  to  all  men,  were  thinges,  openly ,  of  the  world  perceiued.  But  what  rotes 
(otherwife,)vertue  had  faftenedin  his  breft,  what  Rules  of  godly  and  honorable 
k.'  hfe 


John  Dee  his  Mathematical!  P r^face . 

life  he  had  framed  to  him  felfe:  what  vices,  (in  fome  then  liuitig)  notable,  he  tooke 
great  care  to  efchew:  what  manly  vertues ,  in  other  noble  men ,  (  floriihing  before 
his  eyes,) he  Sythingly  afpired  after  :  what  prowefles  he  purpofed  and  menttoa*- 
chieue :  with  what  feats  and  Artes,he  began  to  furnifh  and  fraught  him  felfe ,  for 
the  better  feruice  of  his  Kyng  and  Countrey sboth  in  peace  &  warrc.  Thefe(I  fay) 
his  Heroicall  Meditations  ,  forecaftinges  and  determinations ,  no  twayne ,  (I 
thinke )befide  my  felfe, can  fo  perfe&ly,and  truely  report.  And  therforedn  Con¬ 
fidence,!  count  it  my  part, for  the  honor, preferment,  &  procuring  of  vertue  (thus* 
briefly)  to  haue  put  his  Name ,  in  the  Rcgifter  of  Fame  ImmortalL 

T  o  our  purpofe.  This  lohn,  by  one  of  his  a&es  (bdides  many  other  .'both  in  En¬ 
gland  and  Fraunce,by  me, in  him  noted.  )  did  difclofe  his  harty  loue  to  vertuous 
Scien'ces:and  his  noble intent,to  excell  in  Martiall  proweffe:  When  he,with  hum¬ 
ble  requeft, and  inflantSollicitingrgot  the  bed  Rules  (either  in  time  paft  by  Greke 
or  Romaine,or  in  our  time  vfed:and  new  Stratagemes  therin  deuifed)  for  ordring 
of  all  Companies/ummes  and  N  umbers  of  me, (Many, or  Few)  with  one  kinde  of 
weaponeer  mo,  appointed:with  Artiliery,or  without:on  horfebacke,  or  on  fote: 
to  glue,  or  take  onfet :  to  feem  many,  being  few  :  to  feem  few ,  being  many.  T o 
marche  in  battaile  orlornay :  with  many  fuch  feates,to  Foughten  field,Skarmoufb* 
or  Ambufhe  appartaining:  And  of  all  thefe,liuely  deiignementes  ( moft  curioufly) 
to  be  i n  velame  parchement  deferibed :  with  N otes  &  peculier  markes,as  the  Arte  T{.  M 

requireth :  and  all  thefe  Rules.and  deferiptions  Arithmetical! ,  inclofedina  riche  Earlej'dyed 
Cafe  of  Gold ,  he  vfed  to  weare  about  his  necke :  as  his  Iuell  moft  precious ,  and  Anno.  /  y  y 
Counfaylour  moft  trufty .  Thus,o/r^«z^/^,ofhim,was  ihryned  in  gold :  Of  fkar^e 
Numbers frute , he  had  good  hope.  N ow , N umbers  therfore innumerable , in  l!;”0!?? 
'NumbersytzyfefviS  fhryne  ihall  finde.  fueHhis 

What  nede  I,(forfarder  profe  to  you)  of  the  Scholemafters  of!uftice,to  wife=  Daugh" 
require  teftimony  :how  nedefuil,  how  frutefull ,  how  fkillfull  a  thing  ^Arithmetike  ^ 
is?I  meane,the  Lawyers  ofall fortes.  Vndoubtedly,the  Ciuilians,can  meruaylouf-  merfet. 
ly  declare ; how,neither  the  Auncien;  Romaine  lawes ,  without  good  knowledge 
of  Numbers  art ,cm  be  perceiued :  Nor  (Iuftice  in  Infinite  Cafes)  without  due  pro¬ 
portion,  (narrowly  confidered ,) is  hable  to  be  executed.  How  luftly,  &  with  great 
knowledge  of  Arte, did  PapirJanus  inftitute  a  law  of  partition. ,  and  allowance ,  be- 
twene  man  and  wife  after  a  diuorcec’But  how  Accurfms,  Baldus,Bartalus,Iafon,Alex - 
under ^ and  finally  Alciatus, (being  otherwife,notab!y  Well  learned)do  iumble,geflTe, 
and  erre,from  the  equity, art  and  Intent  of  the  lawmaker :  Arithmetike  can  dete<ft, 
and  conuince:  and  clerely,  make  the  truth  to  fhine.  Good  Bartolus ,  tyred  in  the 
examining  &  proportioning  of  the  matterrand  with  Accurfms  Gloffe,  much  cum- 
bredrburft  o,ut,and  faydiNu/la  ejiin  teto  libro , hacglojfa  diffcilior :  Cuius  computation 
nem nec  Scholafiici nec  Doctor  es intclUgunt.  &c .  Thatis:  Jn  the Svhole  booke ,  there 

is  no  Glojje  harder  then  this :  Whofe  accoumpt  or  reckenyng  3  neither  the  Scho* 
lerspior  the  fdociours  'bnderftand.&c.  What  can  they  fay  of  lulianus  law ,  Si 
ita  Scriptum . efr . O f  th e  Teftators  will  iuftly  performing,  betwene  the  wife ,  Sonne 
and  daughter  <  How  can  they  perceiue  the  aquitie  of  Ayhricanus ,  Arithmetic  all 
Reckening, where  he  treateth  of  LexFaicidiai  How  can  they  deliuer  him, from  his 
Reprouers :  and  their  maintained :  as  Ioannes ,  Accurfms  Hypclitus  and  alciatus? 

How  luftly  and  artificially,  was  Africantis  reckening  madec'Proportionating  to  the 
Sommes  bequeathed, the  C  on  trib  u  tio ns  ofeche  part  Namely, for  the  hundred 
prefently  receiued,i7  ~ .  And  for  the  hundred,  receiued  after  ten  monethes,i2 
A-;  which  make  the  30:  which  were  to  be  cotributed  by  the  legataries  to  the  heire» 

a.j.  For, 


Inflict . 


ST 


j 


lohn  Dee  his  Mathematical!  Preface. 

For,what  proportion^  oo  hath  to  75  :  the  fame  hath  17  JL  to  12  d_ :  Which  is  Sef- 

quitertia:  that  is,as  4, to  3  .which  id&key.  Wonderfull  many  places,  in  theCiuile 
law, require  an  expert  Arithmetics, to  vnderftand  the  deepe  Iudgemet,&  Iuft  de¬ 
terminate  of  the  Auncient  Romaine  Lawmakers .  But  much  more  expert  ought 
he  to  be,  who  iliouid  be  hable  ,  to  deride  with  tequitie, the  infinite  varietie  of 
Cafes, which  do, or  may  happen ,  vnder  euery  one  of  thole  lawes  and  ordinances 
Guile.  Hereby,eafely,ye  may  now  conietiure:  thatin  the  Canon  law:  and  in  the 
lawes  of  the  Realme  (which  with  vs ,  beare  the  chief  Authoritte  ) ,  Tuftice  and  e- 
quity  might  be  greately  preferred,and  fkilfully  executed,  through  due  fkill  of  A- 
rithmetike,and  proportions  appertainy-ng.  The  worthy  Philofbphers ,  and  pru¬ 
dent  Iawmakers(who  haue  written  many  bookes  De  Republica:  How  the  belt  ftate 
of  Common  wealthes  might  be  procured  and  mainteined, )  haue  very  well  deter¬ 
mined  ofluftice :  (which,  not  onely,  is  the  Bale  and  foundation  of  Common 
weales  :but  alfo  the  totall  perfection  of  all  our  workes,  words,  and  thoughtes :  jde- 
fining  it, to  be  that  vertue,by  which, to  euery  onc,is  rendred,  that  to  him  appertai¬ 
ned!.  God  challengeth  this  at  our  handes,to  be  honored  as  God:  tobeloued,as 
a  father :  to  be  feared  asaLord  &  mafter.  O  ur  neighbours  proportions1  alfo  prcf- 
cribed  of  the  Almighty  lawmaker:  which  is  >  to  do  to  other ,  euen  as  we  would  be 
done  vnto.  Thefe  proportions,  are  in  Iuflice  neceflary  :in  duety, commendable: 
and  of  C  ommon  wealthes,  the  life,ftrength ,  flay  and  florilhing.  ^AnjlotL  in  his 
Ethikes  (to  fitch  the  fede  of  Iuflice, and  light  of  diretiion,  to  vfe  and  execute  the 
fame)  was  fayne  to  fly  to  the  perfection,  and  power  of  Numbers :  for  proportions 
Arithmetical!  and  Geometricall.  Plato  in  his  booke  called  Epinomis  ( which  boke, 
is  the  Threafury  ofall  his  dotin' nc)  where, his  purpofe  is, to  feke  a  Science,  which, 
when  a  man  had  it, perfectly  :he  might  feme, and  fo  be, in  d cde JYife .  He, briefly, of 
other  Sciences  difcourfing,findeth  them,  not  hable  to  bring  it  to  paffe :  Butofthe 
Science  of  Numbers, he  fayrh.  lUaypu  numerum  mortalium  generi  dedit,id profeflo  ef 
fciet .  Dxum  atitem  aliquem ,  magi*  quamfortunam ,  ad falutem  no  fir  am,  hoc  munus  nobis 
arbitror  contuhjfe  .  &c .  Nam  ipfum  bonorum  omnium  Author em,  cur  non  maximi  boni, 
PrudcntiA  dico ,  caufam  arbitramuri  T hat  Science pioerely ,K>htch  hath  taught  man* 
kynde  number  ,Jh all  he  able  to  bryng  it  to  paffe.  And }1  thinke}a  certaine  God , 
rather  then fortune pto  hauegiuen  hs  this gft, for  our  blijfe .  Forptohy Jhould 
~tye  not  hedge  him/cho  is  the  Author  of  all  good  things  }to  be  alfo  the  caufe  of the 
greatejlgo'o4t'bjng,name!yJVjfedome ?  There, at  length,he  proueth  Wifedome 
to  be  atteyned ,  by  good  Skill  of  Numbers .  With  which  great  T  eftimony,  and  the 
manifold  profes ,  and  reafons ,  before  exprefled ,  you  may  be  fufficiently  and  fully 
perfiiaded :  of  the  perfect  Science  of  Arithmetike, to  make  this  accounte ;  That  of 
all  Sciences,next  to  Thsolcgie,it  is  mofl  diuine,moft  pure, mofl  ampleand  generall, 
moft  profour.de ,  moft  1  ubtile  ,moft  commodious  and  mofl:  neceflary .  Whole 
nextSifter,is  the  Abfolute  Science  of  Magnitudes:  of  which  (by  the  Diretiion  and 
aide  of  him, whole  (jMagmtude  is  Infinite,and  ofvs  Incomprehenlible  )  I  now  en- 
tend ,  fo  to  write ,  that  both  with  the  ^Multitude, and  alfo  with  the  c Magnitude  of 
MeruaylouS  and  frutefull  verities ,  you  (  my  frendes  and  Countreymen  )  may  be 
ftird  vp,  and  awaked,  to  behold  what  certaine  Artes  and  Sciences,  (to  our  vn~ 
Ipeakabic  behofe)our  heauenly  father,  hath  for  vs  prepared,  and  reuealed,by  liin* 
dry  Philo [others  and  c Mathematiciens . 

ffdth^Number  and  c_ Magnitude ,  haue  a  certaine  Originall  fede,  ( as  it  were  )  of  an 
incredible  property:  and  of  man,  neuer  hable.  Fully,  to  be  declared  .  Of 
Number ,  an  Y nit, and  of  ^Magnitude, a  Poynte,doo  feeme  to  be  much  like  Origi- 
s  Y  nail 


/ 


lohn  Dee  his  Mathematicall  Preface* 

nallcaufes :  But  the  diuerfitie  ncuerthelefle,is  great .  We  defined  an  Vnit ,  to 
be  a  thing  Mathematicall  Indiuifible:  A  Point,  like  wife,  we  layd  to  be  a  Ma¬ 
thematical!  thing  Indiuifible.  Andfarder  ,  that  a  Point  may  haue  a  certaine  de^ 
termined  Situation:  that  is,  that  we  may  afligne,and  prefcribe  a  Point, to  be  here, 
there ,  yonder.  &c.  Herein ,  (behold)  our  Vnit  is  free, and  canabyde  no  bon¬ 
dage,  or  to  be  tyed  to  any  place,or  feat:  diuifible  or  indiuifible .  Agayne ,  by  rea- 
fon,a  Point  may  haue  a  Situation  limited  to  him.-  a  certaine  motion, therfore  (to  a 
place, and  from  a  place)  is  to  a  Point  inciden  t  and  appertainyng.  But  an  Vnit,c an 
not  be  imagined  to  haue  any  motion .  A  Point, by  his  motion,  produceth ,  Ma¬ 
thematically^  line:  (as  we  layd  before)which  is  the  firft  kinde  of  Magnitudes,and 
mod  fimple:  An  Vnit,cm  not  produce  any  number .  A  Line,  though  it  be  produ¬ 
ced  of  a  Point  moued,yet,it  doth  not  confift  of  pointes  :  N  umber ,  though  it  be 
not  produced  of  an  Vnit ,  yet  doth  it  Confift  of vnits ,  as  a  materiall  caufe .  But 
formally,N umber, is  the  Vnion,  and  VnitieofVnits  .  Which  vnyting  and  knit-  Numbed 
ting, is  the  workemanlhip  of  our  minde:  which,of  diftinift  and  difcrete  Vnits ,  ma- 
keth  a  N  umber:  by  vniformitie,refulting  of  a  certaine  multitude  of  Vnits.  And  fo, 
euery  number, may  haue  his  leaft  part,giuem-  namely,  an  Vnit:  But  not  of  a  Magni¬ 
tude,  (no,  not  of  a  Lyne,)the  leaft  part  can  be  giue:bycaufe,infinitly,  diuifion  ther- 
of,may  be  concerned.  All  Magnitude,is  either  a  Line, a  Plaine,  or  a  Solid.  Which 
Line, Plaine, or  Solid, of  no  Senfe,can  be  perceiued,  nor  exactly  by  had  (any  way ) 
reprefented:nor  ofNature  produced:  But,  as  (  by  degrees )  Number  did  come  to 
our  perceiuerance:  So,by  vifible  formes, we  are  holpen  to  imagine,  what  our  Line 
Mathematicall,  is.  What  our  Point,  is.So  precile,.are  our  Magnitudes ,  that  one 
Line  is  no  broader  then  an  other:  for  they  haue  no  bredth  :  Nor  our  Plaines  haue 
any  thicknes.Nor  yet  our  Bodies,any  weight.-be  they  neuer  fo  large  of  dimenfio. 

Our  Body  es,  we  can  haue  Smaller,  then  either  Arte  or  Nature  can  produce  a- 
ny  :  and  Greater  alfo ,  then  all  the  world  can  comprehend .  Our  leaft  Mag¬ 
nitudes,  can  be  diuided  into  fo  many  partes ,  as  the  greateft .  As,  a  Line  of  an 
inch  long,  (with  vs)  may  be  diuided  into  as  many  partes,  as  may  the  diame¬ 
ter  ofthe  whole  world ,  from  Eaft  to  Weft  .-  or  any  way  extended :  What  priui- 
ledges,  aboue  all  manual  Arte,  and  Natures  might,  haue  our  two  Sciences  Ma¬ 
thematically  to  exhibite,afid  to  deale'  with  thinges  offuch  power,  liberty,  fimplici- 
ty,puritie,and  perfe&ionc'  And  in  them,fo  certainly,fo  orderly ,fo  ptecifeiy  to  pro- 
:cede:as,excellentis  that  workema  Mechanical!  Iudged ,  who  nereftcanapproche 
to  the  reprelenting  of workes,  Mathematically  demonftrated  i  And  our  two  Sci¬ 
ences, remaining  pure, and  abfolute,in  their  proper  termes,and  in  their  owne  Mat- 
ter:  to  haue, and  allowe,onely  fuch  Demonftrations ,  as  are  plaine ,  certaine ,  vni- 
tierlall,  and  of  an  seternall  Veritye'This  Science  of  ^Magnitude,  his  properties,con-  Geometric  • 
ditions,and  appertenances :  commonly ,now  is,and  from  the  beginnyng ,  hath  of  * 

all  Philofophers ,  ben  called  Geometric .  But,veryly,with  a  name  to  bafe  andfcant, 
for  a  Science  of fuch  dignitie  and  amplenes.  And,perchaunce ,  that  name, by  co- 
mon  and  fecret  confent,of all  wifemen,  hitherto  hath  ben  fuffred  to  remayne.-that 
it  might  carry  with  it  a  perpetuall  memorye,  ofthe  firft  and  notableft  benefite,  by 
that  Science,  to  common  people  file  wed :  Which  was ,  when  Boundes  and  meres 
of  land  and  ground  were  loft,  and  confoundedfas  in  %y/tf>yearely,with  the  ouer- 
flowyng  of Nilas, the  greateft  and  longeft  riuer  in  the  world  )  or ,  that  ground  be¬ 
queathed,  were  to  be  a(figned:or,  ground  fold,  were  to  be  layd  out :  on  (when  dis¬ 
order  preuailed)that  Commos  were  diftributed  into  feueral  ties.  For,  where,  vpon 
thefe  &  fuch  like  occafios,Some.by  ignorace,  fome  by  negligece,  Some  by  fraude, 
and  fome  by  violence,  did  wrongfully  limite,meafure,  encroach, or  challenge  (  By 

a.ij.  pretence 


lohn  Dec  his  Mathematical!  Preface. 

pretence  ofiuft  content,  and  meafure)  thole  Iandes  and  groundes :  great  lofle4dif- 
quictnes, murder, and  warredidffull  oft)enfue:Till,by  Gods  mercy,and  mans  In- 
duflrie,The  perfect  Science  of  Lines, Plaines,  and  Solides  (like  a  diuine  Iufticicr,) 
gaue  vnto  eiiery  man,  his  owne.  The  people  then,by  this  art  pleaTured,and  great¬ 
ly  relieuedjin  their  Iandes  iuft  mearuring:&  other  Philofophers,  writing  Rules  for 
land  meafuring.  betwene  them  both, thus, confirmed  the  name  of  Geometria, that  is, 
(according  to  the  very  etimologie  of  the  word)Land  meafuring.Wherin,the  peo¬ 
ple  knew  no  farder, of  Magnitudes  vfe,but  in  Plaines:  and  thePbilofophers,ofthe, 
had  no  feet  hearers,  or  Scholers.-farder  to  difclofe  vnto ,  then  of  flat ,  plaine  Geome¬ 
tric.  And  though, thefe  Philofophers,knew  offardervfe,and  bed  vnderflode  the 
etymologye  of  the  worde,yet  this  name  Gcmetria, was  of  them  applyed  generally 
to  all  fortes  of  Magnitudes ;  vnleaft,  otherwhile,  of  Plato ,  and  Pythagoras  .*  When 
KPlatt.  7.  dt  tliey  would  precifely  declare  their  owne  dodrine.  Then, was  *  Geometria ,  with 
"Kip'  xhemftudiumquod circa  planum  verfatur.  But,  well  you  may  perceiue  by  Eudides 

Elementes ,  that  more  ample  is  our  Science ,  then  to  meafure  Plaines:and  nothyng 
lefle  therin  is  toughtf of  purpolejthen  how  to  meafure  Land.  An  other  namc,ther- 
fore,muft  nedes  be  had,  for  our  Mathematical!  Science  of  Magnitudes :  which  re¬ 
garded!  neither  clod, nor  turff:  neither  hill, nor  dale.  neither  earth  nor  heauen;  but 
is  abfolute  CM* egethologia .-not  creping  on  ground ,  and  daffeling  the  eye,  with  pole 
55  perche,rod  or  lyne.-butliftyng  the  hart  aboue  the  heauens,by  inuifibie  lines ,  and 
O*  immortall  beames  meteth  with  the  reflexions, of  the  light  incomprehenfible:  and 
r>  fo  procureth  Ioye,and  perfedion  vnfpeakable.  Of  which  true  vfe  of  our  c Me%e- 
thica,ov  tJl-f egethologia,  Diuine  Plato  feemed  to  haue  good  tafte,and  iudgement.-and 
(by  the  name  ol Geometric  )  fo  noted  it -and  warned  his  Scholers  therof:  as,in  hys 
feuenth  Dialog ,  of  the  Common  wealth,may  euidently  be  fene.  Where  (in  La- 
tin)thus  it  is :  right  well  mandated :  P  r  of eclo, nobis  hoe  non  hegabunt ,  Quicmfy  vclpau. 
litlum  quid  Geometria gufiarunt,  quin  bac  Scientia ,  contra,  omnino fe  habeat ,  quamde  ea 
loquuntur ,  qui  in  ipfa  verfantur .  In  Englifh, thus.  Verely(f ay th  Plato foohofoeuer 

bauef  hut  euen  "Very  litle  flailed  of Geometric, will  not  denye  Tmto  Vs ,  this :  but 
that  this  Science ,is  of  an  other  condicion, quite  contrary  to  thatftohich  they  that 
are  exercifed  in  it ,  do  fpeake  of  it.  And  there  it  followeth,  of  our  Gecmetrie, 
gupd quaritur  cognofcendi  illtus gratia, quod  femper  eft, non  dr  eius  quod  oritur  quandotf 
dr  interit.  Geometria,  eius  quod  eft  femper,  Cognitio  efl.^ttolletigitur{o  Generofe  vir)  ad 
Veritatem^anmum-atfyita^ad  Philofophandum  preparabit  cogitationemjvt  adfopera  con - 
uertamus -quajiuncyontra  quam  decetyid  inferior  a  deijeimus.  dre .  Quam  maximeigitur 
pracipiendum  efL'vt  qui  praclarifsimam  banc  habitat  Civitatem,nullo  modo,Geometriam 
fernant .  Nam  dr  quee  prater ipfms propofitum,quodam  modo  effe  videnturfhaud  exigua 
font.  drc.It  inuft  nedes  be  confdfed  (faith  Plato )  That  £  Geometric}  is  learned  ,  for 

the  knowyng  of  that ,  mhich  is  euer.and not  of  that,  *0 vhicb,in  tymejboth  is  bred 
and  is  brought  to  an  ende.iyc. Geometric  is  the  knowledge  of  that  which  is  euer * 
lajlyng.  It  mill  lift  Vp  therfore( 0  Gentle  Syr  )  ourmynde  to  the  Veritie :  and  by 
that  meanest  mill  prepare  the  T  bought, to  the  Thtlofophicall  loue  ofmifdome: 
that  me  may  turne  or  conuert , toward  heauenly  t  hinges  iSett  mjnde  a*d  thouShf\  mhich 
now ,otherwife  then  becommeth  V>s,me  call  down  on  bafe  or  inferior  things. <zsrc. 
Chiefly,  therfore,  Commaundement  mufl  be  giuen ,  that  ftich  as  do  inhabit  this 
mofl  honorable  Qtie,by  no  meanes,  delpife  Geometric.  For  euen  thofe  thinges 
&>»*  tj  itymhkhjn  manner,  feame  to  be  ,  befide  the purpofe  of  Geometric :  are  of 

m 


iohn  Dee  h  is  Adatfiematfcail  Preface. 

no  frriall  importance .  ^c.  And  befides  the  manifold  vfes  of  Geometrie,  in  matters 
appertainyng  to  warre,he  addeth  more,offecond  vnpurpofed  frute,  and  commo* 
ditye,arrifing  by  Geometrie :  fay  in  g :  Seim  us  quin  etiamyid  Difciplinas  omnes  faciliusper 
difcendas ,1'Mereffe  ommno,atpgerit  ne  G eometriam  altquis,an  non .  &c.  Hanc  ergo  D  o- 
clrinam^fectmdo  loco  difeendam  Imenibus Jlatmmus .  That  is.  (But >alfo  3loe  kno"%>} 

that  for  the  more  eafy  learnyng  ofallyfrtesft  importeth  much ,  whether  one 
haue  any  knowledge  in  Geometrie  }or  no.  <£rc.  Let  las  therfore  make  an  ordi* 
nance  or  decree ,  that  this  Science ,  of  young  men J. hall  be  learned  in  the fecond 
place.  This  was  Diuine  Plato  his  Iudgement,both  of  the  purpofed ,  chief,  and 
perfect  vfe  of  Geometrie:  and  ofhis  fecond, dependyng  ,  deriuatiue  commodities. 

And  for  vs,Chriften  men, a  thoufand  thoufand  mo  occafions  are,  to  haue  nede  of 
the  helpe  of*  CMegethologicall  Contemplations ;  wherby,to  tray  ne  our  Imagina-  *  f  . 
tions  and  Myndes ,by  litle  and  litle,to  forfake  and  abandon,the  groffe  and  cortup- 
tible  Obiedcs,of our  vtward  fenfes.-and  to  apprehend  ,  by  fure  dodrine  demon- 
ftratiue,Things  Mathematical!.  And  by  them ,  readily  to  be  holpen  and  con- ' earthly  name. t 
duCtcd  to  conceiue ,  difeourfe  ,  and  conclude  of  things  Intellectual  ,  Spiritual!,  of  Geometrie* 
£temall,and  fuch  as  concerneour  Bliffe  euerlafting ;  which,  otherwise  (  without 
Special!  priuiledge  of  Illumination,  or  Reuelation  fro  heauen  )  No  mortall  mans 
wyt( naturally)  is  hable  to  reach  vnto,or  to  Compare.  And,veryly,by  my  fmall 
T aient(from  aboue)I  am  hable  to  proue  and  teftifie,that  the  litterall  T ext,and  or¬ 
der  of  our  diuine  Law,Oracles,ana  Myfteries,require  more  fkill  in  Numbers, and 
Magnitudes  .•  then  (commonly)  the  expofitors  haue  vttered :  but  rather  onely  (at 
the  moftjfo  warned  :  &  (hewed  their  own  want  therin. (To  name  any,  is  nedeles: 
and  to  note  the  places, is, here, no  place:  But  if  I  be  duelyafked,my  anfwere  is  rea¬ 
dy.)  x^nd  without  the  litterall,Grammaticall,Mathematicall  or  Naturali  verities  of 
fuch  places  ,  by  good  and  certaine  Arte,perceiued,no  Spirituall  fenfe  ( propre  to 
thofe  places, by  Abfolute  T heologie) will  thereon  depend.  N o  man, therfore,  can  ^ 

doute  3  but  toward  the  atteyning  of  knowledge  incomparable  ,  andHeauenly 
Wifedome.*  Mathematicall  Speculations, both  ofNumbers  and  Magnitudes:  are  ” 
meanes, .aydes,  and  guides: ready,  certaine  , and  neceflary.  From  henceforth,in 
this  my  Preface, will  I  frame  my  talke,to  Plato  his  fugitiue  Scholers:  or,  rather  ,  to 
fuch,  who  well  can,( and  alfo  wil,)vfe  their  vtward  fenfes,to  the  glory  of  God,the 
benerite  of  their  Coun  trey, and  their  ownefecretcontentation,  or  honeft  prefer¬ 
ment,  on  this  earthly  Scaffold.  T o  them,I  will  orderly  recite,  deferibe  &  declare 
a  great  Number  of  Artes ,  from  our  two  Mathematicall  fountaines ,  deriued  into 
the  fieldes  of  Nature.  Wherby ,  fuch  Sedes ,  and  Rotes ,  as  lye  depe  hyd  in'the 
groud  of  ‘Nature, are  refrefhed, quickened, and  prouoked  to  grow,  (bote  vp,  fioure, 
and  giue  frute, infinite,and  incredible.  And  thefe  Artes,fhalbe  fuch ,  as  vpon  Mag¬ 
nitudes  properties  do  depen de,more,then  vpon  N umber.  And  by  good  reafon 
we  may  call  them  Artes,and  Artes  Mathematicall  Deriuatiue :  for  (  at  this  tyme)I  *AnArtu 
Define  An  Arte, to  be  a  Methodicall  coplete  Dodfrine, hailing  abun- 
dancy  of  fufhcient,and  pearlier  matter  to  deale  with, by  the  allow¬ 
ance  of  the  Metaphificall  Philofopher  :  the  knowledge  whereof,  to 
humaine  ftate  is  neceflarye.  And  that  I  account,  An  Art  Mathemati-  °*rt  Math** 
call deriuatiue,  which  by  Mathematicall  demonftratiue  Method,  TuTtiue^ 
in  Nubers ,  or  Magnitudes, ordiah  and  confirmeth  his  dodtrine,  as 
much  &  as  perfedtly ,  as  the  matter  fubiedt  will  admit .  And  for  that, 

a.iij,  I  emend 


A  Mechani- 
tietu 


I. 


Geometric 
vulgar . 


z. 


1 . 


i. 


Note, 


Note, 


lohn  Dee  his  Mathematical!  Preface, 

I  entend  to  vfe  the  name  and  propertie  of  a  Mechanicien,  o  therwife,th  en  (hi  th  er  to) 
it  hath  ben  vfedj  thinke  it  good,  (for  diftin&ion  fake)  to  giue  you  alfo  a  brief det 
cription,  what  I  meane  therby.  A  Mechanicien,or  a  Mechanicali  work¬ 
man  is  he ,  whofe  f  kill  is ,  without  knowledge  of  Mathematical! 
demonftration ,  perfectly  to  worke  and  finifhe  any  fenfible  worke, 
by  thd  Mathematicien  principall  or  deriuatiue,  demonflrated  or  de- 

monilrabie.  Full  well  I  know, that  he  which  inucnteth,  or  maketh  thefe  de- 
monftrations,is  generally  called  ffeculatiue  CMechanicien :  which  differreth  no- 
thyng  from  a  Mechanicali  'jMatkematicicn .  So, in  refpcd  of  diuerfe adtions,one 
man  may  haue  the  name  offundry  artes:as,fome  tyme,ofa  Logicien ,  fome  tymes 
(in  the  fame  matter  otherwife  handled)  of  a  Rethoricien .  Of  thefe  trifles,I  make, 
(asnow,in  refped  of  my  Preface, )fmall  account:  to  fyle  the  for  the  fine  handlyng 
offubtile  curious  difputers .  In  other  places ,  they  may  commaunde  me, to  giue 
good  reafon :  and  yet, here, I  will  not  be  vnreafonable. 

Firft, then, from  the  puritie,abfolutenes,and  Immaterialitie  of  Principall  Geo¬ 
metric,  is  that  kindc  of  Geometric  deriued ,  which  vulgarly  is  counted  Geometric  : 
and  is  the  Arte  of  Meafuring  fenfible  magnitudes,  their  i'ufft  quatities 
and  contentes  .  This,  teacheth  to  meafure,either  at  hand:  and  the  praftifer,  to 
be  by  the  thing  Meafured.*  and  fo,by  due  applying  of  Cumpafe,  Rule,  Squire, 
Yarde,Ell,Perch,Pole,Line,Gagingrod,(or  filch  like  inftrument) to  the  Length, 
Plaine,or  Solide  meafured,  '‘to  be  certified,  either  of  the  length,  perimetry,  or  di- 
fiance  lineall :  and  this  is  called, UMecometrie .  Or*  to  be  certified  of  the  content  of 
any  plaine  Superficies :  whether  it  be  in  ground  Surueyed,  Borde,  or  Glafle  mea- 
fiired,or  fuch  like  thing :  which  meafuring,is  named  Embadometrie .  *Or  els  to  vn- 
derfland  the  Soliditie,and  content  of  any  bodily  thing :  as  ofTymber  and  Stone, 
or  the  content  ofPits,Pondes,Wells,Veffels,fmaU&  great,of  all  fafhions.Where, 
ofWine,Oyle,Beere,or  Ale  veffells,&c,the  Meafuring-commanly,  hath  a  pecu- 
lier  name.-and  is  called  Gaging .  And  the  generallname  ofthefe  Solide  meafures, 
is  Stereometric .  Or  els, this  vulgar  Geometric ,  hath  confideration  to  teach  the  prac- 
tifer ,  how  to  meafure  things, with  good  diflance  betwene  him  and  the  thing  mea¬ 
fured  :  and  to  vnderfland  thereby,either  *how  Farre,athingfeene(on  land  or  wa¬ 
ter)  is  from  the  meafurer:  and  this  may  be  called  Jfornecometrie:  Or,how  High  or 
depe,aboue  or  vnder  theieuel  of  the  meafurers  ftading,any  thing  is, which  is  fene 
on  land  or  water,  called  Hypfometrie.*Qi it  informeth  the  meafurer ,  how  Broad 
any  thing  is, which  is  in  the  meafurers  vew:fo‘itbc  on  Land  or  Water,fituated:and 
may  be  called  Plat ometrie .  Though  I  vfe  here  to  condition,the  thing  meafured,  to. 
be  on  Land,  or  Water  Situated :  yet,  know  for  ceitaine,  that  the  fundry  heigthe  of 
Cloudes,  blafing  Starres,  and  of  the  Mone  ,may(by  thefe  meanes)haue  their  di- 
flances  from  the  earth  :  and,  of  the  blafing  Starresand  Mone,the  Soliditie  (afwell 
as  difiances) to  be  meafured:But  becaufe, neither  thefe  things  are  vulgarly  taught: 
nor  of  a  common  praftifer  fo  ready  to  be  executed  V  I,rather,let  fuch  meafures  be 
reckened  incident  to  fome  of  our  other  Artes,  dealing  with  thinges  on  high,more 
purpofely,  then  this  vulgar  Land  meafuring  Geometrie  doth :  as  in  Perjpetfiue  and 
t^AUronomie,  &c .  f 

QF  . thefe  feates  (  Farther  applied  )  is  Sprongthe  feateof  Geodejie  ,  or  Land 
Meafuring:  more  cunningly  to  meafure  &  Suruey  Land,  Woods,  and  Waters, 
a  farre  of.  More  cunningly, I  fay :'  But  God  knoweth  (hitherto)  in  thefe  Realmes 
of  England  and  Ireland  (  whether  through  ignorance  or  fraude ,  I  can  not  tell ,  in 
e,uery  particular )  how  great  wrong  and  iniurie  hath  (in  my  timejbene  committed 


lohn  Dee  His  Mathematical!  Preface, 

by  vntrue  meafuring  and  furueying  ofLand  or  Woods, any  way  .  And,  this  I  art! 
fure:  that  die  Value  of  the  difference,  bet  wene  the  truth  and  fuch  Suru  eyes, would 
haue  bene  hable  to  haue  loud  (for  euer)  in  eche  of  our  wo  Vniuerfitics,an  excel¬ 
lent  Mathematicall  Reader:  to  eche,allowing  (yearly)  a  hundred  Markes  oflawfull 
money  of  this  realme:  which, in  dede,would  feme  requifit,here,to  be  had  (though 
by  other  wayes  prouided  for)  as  well,as,the  famous  Vniuerfitie  of  Paris,  hath  two 
Mathematicall  Readers  :  and  eche, two  hundreth  French  Crownes  yearly,  of  the 
French  Kinges  magnificent liberalitie  onely »  Now,againe,  to  our  purpofe  retur¬ 
ning  :  Moreouer,  of  the  former  knowledge  Geometricall,aregrowen  the  Skills  of 
Geographic  ,  Chorographie  ,  Hydrographic  ,  and  Stratarithmetrie . 

Geographic  teacheth  wayes, by  which,  in  fudry  formes, (as  Sph<zrike,Vlaine  n 
or  other)  ,the  Situation  of  Cities,  Townes,Villages,  Fortes,CaftelIs,Mountaines,  „ 
Woods,Hauens,Riuers,Crekes,&  fuch  other  things,vpo  the  outface  of  the  earth-  „ 
ly  Globe  (either  in  the  whole,or  in  lome  principall  meter  and  portion  therofco-  „ 
tayned)may  be  defcribed  anddefigned, in  comenlurations  Analogicall  to  Nature 
and  veritierand  moft  aptly  to  our  vew,may  be  reprefented.Of  this  Arte  how  great  7f 
pleafure,and  how  manifolde  commodities  do  come  vnto  vs,daily  and  hourely :  of 
moft  men,  is  perceaued .  While, fome,  to  beatitifie  their  Halls,Parlers,  Chambers, 
Galeries,Studies,or  Libraries  with:  other  fome,for  thinges  paft,  as  battels  fought, 
earthquakes, heauenly  fyringes,&fuch  occurentes,in  hiftories  mentioned:  therby 
liuely  ,as  it  were, to  vew  e  the  place,the  region  adioyning,the  diftance  from  vs :  and 
fuch  other  circumftances .  Some  other, prefently  to  vewe  the  large  dominion  of 
theTurke  :  the  wide  Empire  of  the  Mofchouite:  and  thelitle  morfell  of  ground, 
where  Chriftendome(by  profeffion)is  certainly  knowcn.  Litle,Ifay,in  rclpe&e  of 
the  reft,  &c.  Some, either  for  their  owne  iorneyes  direding  into  farre  landcs: 
or  to  vnderftand  of  other  mens  trauailes .  To  conclude,  fome,  for  one  purpofe ; 
and  fome, for  an  other,  liketh,loueth,getteth,and  vfeth,  Mappes,  Chartes,&  Geo* 
graphical!  Globes .  Ofwhofe  vie,  to  fpeake  fuff  ciently,  would  require  a booke 
peculier. 

Chorographie  feemeth  to  be  art  vnderling,  and  a  twig,  of  Geographic: 
and  y  et  neuerthelefte,  is  in  pradile  manifolde,  and  in  vfe  very  ample .  This  tea-  ,5 
cheth  Analogically  to  defcribe  a  fmall  portion  or  circuite  of  ground,  with  the  con-  „ 
rentes  :  not  regarding  what  commenfuration  it  hath  to  the  wholes,  or  any  parcell,  „ 
without  it,  contained .  Butin  the  territory  or  parcell  of  ground  which  it  taketh  in  » 
hand  to  make  defcription  of,  itleaueth  out  (orvndefcnbed)  no  notable ,  or  odde  „ 
thing,  aboue  the  ground  vifible  .Yea  and  fometimes ,  of  thinges  vnder  ground,  ,, 
geueth  fome  peculier  marke  .*  or  warning :  as  ofMettall  mines,  Cole  pittes,  Stone  „ 
quarries.  &c.  Thus,  a  Dukedome,a  Shiere,a  Lordfliip,  or  Idle,  may  be  defcribed  „ 
diftindly .  But  marueilous  pleafant,  and  profitable  it  is ,  in  the  exhibiting  to  our 
eye, and  comhienfuration,  the  plat  of  a  Citie,  Towne,  Forte,  or  Pallace,  in  true 
Symmetry :  notapproching  to  any  of  them :  and  out  of  Gunne  fhot.&c.  Hereby, 
the  K^frchitett  may  furnifhe  him  felfe, with  ftore  of  what  patterns  he  liketh  :  to  his 
great  inftrudion:  euen  in  thofe  thinges  which  outwardly  are  proportioned: either 
limply  in  them  felues :  or  refpediuely,to  Hilles,Riuers,  Hauens,  and  Woods  ad- 
ioyning .  Some  alfo,  terme  this  particular  defcription  of  places ,  Topographic . 

HydrOgraphlC,deliuereth  to  our  knowledge ,  on  Globe  or  inPlaine,  „ 
the  peifed  Analogicall  defcription  of  the  Ocean  Sea  coaftes,  through  the  whole 
world  :  or  in  the  chiefe  and  principall  partes  thereof :  with  the  lies  and  chiefe  sj 

adfij*  paticular 


*Nate', 

Thedijfe-  ,, 
rence  be-  „ 
ttyene  Stra-  „ 
tarithme -  ,, 
trie  and.  3J 
Tafiicie, 


lohn  Dee  his  Mathematical!  Preface. 

particular  places  ofdaungers,  conteyned  within  the  boundes.,and  Sea coafteS  de- 
icribed :  as,  of  Quicldandes,Bankes-,Pittes,Rockes,Races,CountertideSiWhorle» 
pooles.  &C.  This,  dealeth  with  the  Element  of  the  water  chiefly ;  as  Geographic, 
did  principally  take  the  Element  of  the  Earthes  defcription  (  with  his  apperte- 
nances  )  to  taske  .  And  befides  thys  ,  Hyd.rographie  ,  requireth  a  particular 
Regifter  of  certaine  Landmarkes  (where  markes  may  be  had)  from  the  fea,well  lia¬ 
ble  to  be  fkried,  in  what  point  of  the  Seacumpafe  they  appeare,and  what  apparent 
form  e,S.ituation,  and  bignes  they  haue,  in  refpe&e  of  any  daungerous  place  in  the 
fea,or  nere  vnto  it,  affigned:  And  in  all  Coaftes,  what Morte,maketh  full  Sea.-and 
what  way,  the  Tides  and  Ebbes,  come  and  go, the  Hydrographer  oughtto  recorde. 
The  Saundinges  likewife :  and  the  Chanels  wayes:  their  number, and  depthes  or¬ 
dinarily,  at  ebbe  and  flud,  ought  the  Hydrographer ,  by  obferuation  and  diligence 
of  Measuring,  to  haue  certainly  knowen .  And  many  other  pointes,are  belonging 
to  perfede  Hydrographies  and  for  to  make  a  Rutter,  by :  of  which,I  nede  not  here 
fpeake  :  as  of  the  defcribing,in  any  place,  vpon  Globe  or  Plaine,  the  ^a.poin’tes  of 
the  Compafe,truely :  (wherof,  fcarflyfoure, in  England ,  haue  right  knowledge: 
bycaufe,  the  lines  thcrof,  are  no  (iraight  lines ,  nor  Circles . )  Of  making  due  pro- 
iedion  of  a  Sphere  in  plaine.Of  the  Variacion  of the  Compas ,  from  true  N  orthe: 
And  fuch  like  matters  (of  great  importance ,  all )  I  leaue  to  fpeake  of  in  this  place: 
bycaufe, I  may  feame(al  ready)to  haue  enlarged  the  boundes,and  duety  of  an  Hy- 
dographer,  much  more,then  any  man  (to  this  day)hath  noted, or  preferibed .  Yet 
am  I  well  hable  to  proue,all  thefe  thinges ,  to  appertaine ,  and  alfo  to  be  proper  to 
the  Hydrographer.  The  chief  vfe  and  ende  of  this  Art,  is  the  Art  ofNauigation: 
butit  hath  other  diuerfe  vies :  euen  by  them  to  be  enioyed ,  that  neuerlacke  fight 
of  land. 

Stratanthmetrie,  is  the  Skill,  (appertainyng  to  the  warre , )  by  which  a 
.man  can  fee  in  figure,analogicall  to  any  Geometrical  figure  appointed,  any  certaine 
number  orfumme  of  men:  offuch  a  figure  capable:  (by  reafon  ofthe  vfuall  (paces 
betwene  Souldiers  allowed :  and  for  that ,  of  men,can  be  made  no  Fradions.  Yet, 
neuertheles,he  can  order  the  giuen  fumme  of  men  ,  for  the  greateftfuch  figure, 
that  of  them,  cabe  ordred)and  certifie,of  the  ouerplus:  (if  any  be)  and  of  the  next 
certaine  fumme, which, with  the  ouerplus,will  admitafigureexadly  proportionail 
to  the  figure  affigned.  By  which  Skill,alfo,ofany  army  or  company  of  men :  (the 
figure  &  fides  of  whole  orderly  (landing, or  array, is  knowen)he  is  able  tp  exprefle 
the  iufl  number  of  men,  within  that  figure  coriteined:  or(orderly  )  able  to  be  con- 
teined.  *  And  this  figure, and  (ides  therof,  he  is  hable  to  know :  either  beyngby, 
and  at  hand:  or  a  farre  of.  Thus  farre, ftretcheth  the  defcription  and  property  of 
Stratarithmetrie :  furficienr  for  this  ty me  and  place .  It  difterreth  from  the  Feate 
Ta£licall,De  dciebus  injlruendd.  bycaufe,  there,  is  neceffary  the  wifedome  and  fore¬ 
fight, to  what  purpofe  he  fo  ordreth  the  men ;  and  Skillfull  liability ,  alfo ,  for  any 
occafion,or  purpofe ,  to  deuife  and v(e  the  apteft  and  mod  neceflary  order',  array 
and  figure  of  his  Company  and  Summe  of  men  .  By  figured  meane.-  as,either  ofa 
Perfect  Square,  Triangle ,  Circle ,  Ouale ,  longfquarc ,  (of  die  Grekes  it  is  called  Eiero - 
rmkes  )  Rhombe,  Rhomboid,  Lunular ,  Ryng,  Serpentine,  and  fuch  other  Geometricall 
figures:  Which,inwarrcs, haue  ben,  and  are  to  be  vfed  :  for  commodioufhes ,  ne- 
ceflity,and  auauntage  &c.  And  no  (mail  f kill  ought  he  to  haue ,  that  lhould  make 
true  reporter  nere  the  truth,of  the  numbers  and  Summes,offootemen  or  horfe- 
men ,  in  the  Enemyes  ordring .  A  farre  of,  to  make  an  eftimate ,  betwene  nere 
termes  of  More  and  Leffe,is  not  a  thyng  very  rife ,  among  thofe  that  gladly  would 


John  Dee  his  Mathematical  Preface. 

do  it.  Great  pollicy  may  be  vfed  of  the  Capitaines,(ar  tymes  fete, and  in  places  t.fil 

conuenientjas  to  vfe  Figures ,  which  make greateft  fhew ,  offo  many  as  hehath:  ou  ^fn‘ie 
an  d  vfing  the  aduauntage  of  the  three  kindes  of  vfuall  fpaces  r  (  betwene  footemen  it  hard, to  perforin 
or  horfemen)to  take  the  largefbor  when  he  would  feme  to  haue  few,  (beyng  ma- 
ny.-  )  contrary  wife, in  Figure, and  Ipace.  The  Herald, Purfeuant,  Sergeant  Royally  ^Zheyl^ifi 
Capitaine ,  or  who  foeuer  is  carefull  to  come  nere  the  truth  herein  ,  befides  the 
Iudgement  ofhis  expert  eye,his  fkill  of  Ordering  TacUcall,  the  helpe  of  his  Geo-  Sides  {ink  Angles J 
metricall  inuru men t : Ring,  or  Staffe  Aftronomicall :  (  commodioufly  framed  for  •And  where.  Refold 
cariage  and  vfe)  He  may  wonderfully  helpe  him  ftlfe*  byperfpediue  Glaffts.In 
which,  (I  truft )  our  pofterity  will  proue  more  fkillfull  and  expert ,  and  to  greater  ^ITg^utdaL. 
purpofes,  then  in  thefe  dayes,  can  (almoft  )be  credited  to  be  poflible.  generally  with  *A- 

Thus  haue  I  lightly  pafied  otter  the  Artificiall  Feates,chiefly  dependyng  Vpoil  and.thatfor  Bat - 
vulgar  Geometric :  &  commonly  and  generally  reckencd  vnder  the  name  of  Geome- 
trie.  But  there  are  other(very  many)  AMethodicall  Artes  j  which,  declyning  from 
the  purity ,  •implicitie,and  Immateriality, of  our  Principall  Science  of Magnitudes:  CA  ' 

do  yet  neuertheles  vfe  the  great  ayde  ,  direction  ,  and  Method  of  the  fayd 
principall  Science ,  and  haue  propre  names ,  and  diftind :  both  from  the  Science 
of  Geometric,  (from  which  they  are  deriued)and  one  from  the  other.  As  Per- 

fpeediue,  Aftronomie ,  Muhke,  Cofmographie,  Aftrologie,Statike, 
Antbropographie^rochilike;,  Helicofophie,  Pneumatithmie,  Me- 
nadrie,  Hypogeiodie,  Hydragogie,  Horometrie,  Zographie,  Archi¬ 
tecture,  Nauigation ,  Thaumaturgike  and  Archemaftrie.  I  thinke  it 
neceflaiy ,  orderly ,  of  thefe  to  giue  fome  peculier  deferiptions  :  andwithall,  to 
touch  fome  of  their  commodious  vfes ,  and  fo  to  make  this  Preface ,  to  be  a  litde 
fwete,pleafant  N ofegaye  for  you.-to  comfort  your  Spirites ,  beyng  almoft  out  of 
courage,  andindefpayre,  ( through  brutifh  brute  )  Weenyng  that  Geometric ,had 
but  ferued  for  buildyng  of  an  houfe,or  a  curious  bridge, or  the  roufe  of  Weftmin- 
fter  hall ,  or  fome  witty  pretty  deuife ,  dr  engyn  ,  appropriate  to  a  Carpenter,or  a 
loyner  &c.That  the  thing  is  farre  otherwife ,  then  the  world ,  (commonly) to  this 
day, hath  demed,by  worde  and  worke ,  good  profe  wilbe  made. 

Among  theft  Artes,  by  good  reafon,P  erlpectiuc  oughtto  be  had ,  ere 
of  l. Aftronomicall  Apparences ,  perfed  knowledge  can  be  atteyned.  And  bycaufe 
of  the  prerogatiue  of  Light ,  beyng  the  firftof  Gods  Creatures:  and  the  eye,  the  light 
of  our  body,  and  hisSenfe  moft  mighty, and  his  organ  moft  Artificiall  and  Geome- 
tricall: At  Pe'rpffiue, we  will  begyn  therfore.  Perfpediue,is  an  Art  Mathe¬ 
matical!, which  demonftrateth  the  maner^and  properties,  of  all  Ra¬ 
diations  DireCt, Broken, and  Reflected. This  Defcription,or Notation ,  is 
briefbut  it  reacheth  fo  farre,as  the  world  is  wyde.  It  concerneth  all  Creatures, 
all  Adions ,  and  palfions,  by  Emanation  of  bearnes  perfourmed .  Beames,or  na- 
turalllines ,  (here)  I  meane ,  notoflight  onely,or  of  colour  (though  they,to  eye, 
giue  fhew, witnes, and  profe ,  wherby  to  ground  the  Arte  vpon  )but  alfo  of  other 
F ormes, both Subjlantiall,  and  Accidentally  the  certain e  and  determined  adiue  Ra« 
diall  emanations.  By  this  Art(omitting  to  fpeake  ofthehigheft  pointes)  we  may 
vfe  our  eyes, and  the  light,with  greater  pleafure:and  perfeder  Iudgementrboth  of 
things, in  light  feen,&  of  other:  which  by  like  order  of  Lightes  Radiations,  worke 
and  produce  their  effedes .  We  may  be  afhamed  to  be  ignorant  of  the  caufe,why 
fo  fundry  wayes  our  eye  is  deceiued,and  abufedras,  while  the  eye  weeneth  a  roud 
Globe  or  Sphere(beyng  farre  of)  to  be  a  flat  and  plain e  Circle,and  fo  likewife  iud- 

b.j.  geth 


<sV*T* 

■.iWwr 


•v.t,  -- 


A  maruciloHs 
Clap.  (TJ3 


s.mp. 


John  Dee  his  Mathematical!  Preface. 

geth  a  plaine  Square,  to  be  roud : fuppofeth  walles  parallels,to  approche,a  farre  of*: 
rofe  and  floure  parallels,the  one  to  bend  downward  ,  the  other  to  rife  vpward,at  a 
little  diftance  from  you.  Againe ,  of  thinges  being  in  like  fwiftnes  of  mouing ,  to 
thinke  the  nerer,ta  moue  faftenand  the  farder,much  flower.Nay,  of  two  thinges, 
wherof the  one  (incomparably)  doth  moue  fwifter  then  the  other  ,  to  deme  the 
flower  to  moue  very  fwift,&  the  other  to  ftand:  what  an  error  is  this,ofour  eye?  Or 
the  Raynbow,  both  of  his  Colours, of  the  order  of  the  colours, of  the  bignes  of  it, 
the  place  and  heith  ofit,(&c)to  know  the  caufes  demonflratiue,is  it  not  pleafant, 
is  it  not  necdlaryc'of  two  or  three  Sonnes  appearing:  ofBlafing  Sterres :  and  fuch 
like  thinges  :  by  naturall  caufes ,  brought  to  paffe ,  (and  yet  ncuertheles ,  offarder 
matter,  Significatiue  )  is  it  not  commodious  for  man  to  know  the  veiy  true  caufe, 
&  occafion  Natural!  i  Yea,rather,is  it  not, greatly,  againft  the  Souerainty  of  Mans 
nature ,  to  be  fo  ouerfhot  and  abufed ,  with  thinges  (  at  hand  )  before  his  eyes  { 
as  with  a  Pecockes  tayle ,  and  a  Doues  necke  :  or  a  whole  ore,  in  water,  hob 
den, to  feme  broken  .  *  Thynges, farre  of, to  feeme  nere :  and  nere,  to  feme 
farre  of  .  Small  thinges  ,  to  feme  great  :  and  great ,  to  feme  final!  .  One 
man,  to  feme  an  Army  .  Or  a  man  to  be  curftly  affrayed  of  his  owne  (had- 
do  w  .  Yea  ,fo  much, to  feare,that,if  you,being(alone )  nere  a  certaine  glaffe ,  and 
proifer,with  dagger  or  fword,to  foyne  at  the  glafle ,  you  fhall  fuddenly  be  irioued 
to  gitiebacke(inmaner)  by  reafon  of  an  Image,  appearing  in  the  ayre,betwene 
you  &  the  glafle,  with  like  hand,  /word  or  dagger,&  with  like  quicknes ,  foyningat 
your  very  eye,  likewife  as  you  do  at  the  Glafle.  Straunge,this  is, to  heare  of:  but 
more  meruailous  to  behold,  then  thefe  my  wordes  can  fignifie.  And  neuerthe- 
lefle  by  demonfiration  Opticall,  the  order  and  caufe  therof,  is  certified:  euen  fo,as 
the  effe<fl  is  confequent.  Y ea,thus  much  more, dare  I  take  vpon  me, toward  the  fa- 
tiilying  of  the  noble  courrage,  that  longeth  ardently  for  the  wifedome  of  Caufes 
Naturalhas  to  let  him  vnderfiand,  that, in  London ,  he  may  with  his  owne  eyes, 
haue  profe  of that,  which  I  haue  layd  herein .  A  Gen  deman,  (which,for  his  good 
feruice,  doneto  his  Countrey,is  famous  and  honorable :  andforfkill  in  the  Ma¬ 
thematical!  Sciences,  and  Languages, is  the  Od  man  of  this  land.  &c. )  euen  he,is 
hable:and(I  am  furc)will,  very  willingly, let  the  Glafle,  and  profe  be  fene.-andfo  I 
(here)  requeft  him  :  for  the  encreafe  of  wifedome ,  in  the  honorable  :  and  for  the 
flopping  of  the  mouthes  malicious :  and  reprdfing  the  arrogancy  of  the  ignorant. 
Y  e  may  eafily  gefle ,  what  I  mcane.  This  Art  of  Perjpmm r,  is  of  that  excellency, 
and  may  be  led, to  the  certifying, and  executingoffuch  thinges ,  as  no  man  would 
eafily  beleue:  without  Adtuall  profe  perceiued.  I  fpeake  nothing  of  Naturall  Phi- 
tefopbie, which,' without  Perjj’cctive,  can  not  be  fully  vnderftanded ,  nor  perfectly  at- 
teined  vnto.  N  or,  of  Jjlronomie:  which, without  P  erjpecHue  ^czn  not  well  be  groun¬ 
ded  :  N  or  '^Afirdogic ,  naturally  Verified,  and  auouched.  That  part  hereof,which 
dealeth  with  Glaflcs(which  name,Glafle,is  a  generall  name,  in  this  Arte ,  for  any 
thing, from  which, a  Beame  reboundeth)  is  called  Catoptrike  >  and  hath  fo  many  v- 
fes,both  merueiloiis,and  proffitable:  that, both, it  would  hold  me  to  long ,  to  note 
therin  the  principall  conclufions,all  ready  knowne:  And  alfo(perchaunce)  fome 
thinges,mightlackedue  credite  with  you :  And  I,  therby,  to  leefe  my  labor:and 
g^.  you, to  flip  into  light Iudgement*, Before  you  haue  learned  fufficiently  thepowre 
of  N ature  and  Arte. 

IMow ,  to  procede:  «AftronoiTllC,is  an  Arte  Mathematical!, which 
demonftrateth  the  difbmce ,  magnitudes ,  and  all  naturall  motions, 
apparences,and  pafsions  propre  to  the  Planets  and  fixed  Sterres :  for 

" 1  any 


IohnDee  his  Mathematical  Preface** 

^any  time  paft, prefen  t  and  to  come:in  relpecf  of  a  certaine  Horizon  i 

or  without  reipect  of  any  Horizon.  By  this  Arte  we  are  certified  of  the  di- 
ftance  of  the  Starry  Skye,and  of  eche  Planete  from  the  Centre  of  the  Earth.-  and  of 
the  greatnes  of  any  Fixed  ftarre  fene,  or  Planete, i  n  reiped  of  the  Earthes  greatnes. 
As ,  we  are  lure  (  by  this  Arte )  that  the  Solidity ,  Mailines  and  Body  of  the  Sonne, 
conteineth  the  quantitieofthe  whole  Earth  arid  Sea, a  hundred  threfcoreand 
two  times ,  leffe  by  ~  one  eight  parte  of  the  earth.  But  the  Body  of  the  whole 
earthly  globe  and  Sea, is  bigger  then  the  body  of  the  Mone  ,  three  and  forty  times 
lefie  by  --  of  the  Mone.  Wherfore  the  Sonne  is  bigger  then  the  CVtone  ,  7000 
times,  idle,  by  55?  C-  that  is ,  precifely  69 40  11  bigger  then  the  CMone.  And  yet 

the  vnfkillfuli  man, Would  iudge  them  a  like  bigge .  Wherfore,of  Necefsity,the 
one  is  much  farder  from  vs, then  the  other.  The  Sonne ,  when  he  is  fardeft  from 
the  earth  (which, now, in  our  age, is, when  he  is  in  the  8  .degree,of  Cancer)is ,  1179 
Semidiameters  of  the  Earth, diftante .  And  the  c M one  when  file  is  fardeft  from  the 
earth, is  68  Semidiameters  ofthe  earth  and  —  The  nereft ,  that  the  CMone  com- 
meth  to  the  earth, is  Semidiameters  52  ~~  The  didance  ofthe  Starry  Skye  is  ,fro 
vs, in  Semidiameters  of  the  earth  20081  ~C'  Twenty  thoufand  fourefcore ,  one, 

andalmoftahaifc.  Subtract  from  this, the  CM  ones  nereft  diftance,from  the  Earth: 
and  therof  remaineth  Semidiameters  of  the  earth  200.29  _l  Twenty  thoufand 

nine  and  twenty  and  a  quarter.  So  thicke  is  the  heauenly  Palace  ,  that  the  Pla~ 
netes  haue  all  their  exercife  in, and  moft  meruailoufly  perfourme  the  Commaude- 
inent  and  Charge  to  them  giuen  by  the  omnipotent  Maieftie  of  the  king  of  kings. 
This  is  that,  which  in  Genefis  is  called  Ha  Rakia .  Confident  well.  The  Semidia- 
meterof  the  earth,  coteineth  of  our  common  miles  3436-1.  three  thoufand, foure 

hundred  thirty  fix  and  foure  eleuenth  partes  of  one  myle.-Such  as  the  whole  earth 
and  Sea,  round  about,  is  21600.  Oneand  twenty  thoufand  fix  hundred  of  our 
myles.  Allowyng  for  euery  degree  of thegreateft  circle, thre  fcore  myles.  Now  if 
you  way  well  with  your  felfe  butthislitleparcell  offrute  ^flronomcall,  as  con¬ 
cerning  the  bignefte,Diftances  of  Sonne, Mone,  Sterry  Sky, and  the  huge  maffinesof 
Ha  Rakia ,  will  you  not  finde  your  Confciences  moued ,  with  the  kingly  Prophet, 
to  fingthe  confeffion  of  Gods  Glory, and  fay,  TheHeauens  declare  the  gW* 

ry  ofGodydncl  the  Firmament  R«ki«\ Jbeweth  forth  the  1 vork.es  of  his  handes. 
And  fo  forth, for  thofe  fiue  firft  ftaues,of  that  kingly  Pfalme.  Well,well,It  is  time 
for  fome  to  lay  hold  on  wifedome, and  to  Iudge  truly  of  thinges:  and  notfo  to  ex¬ 
pound  the  Holy  word,all  by  Allegories  :  as  to  Negleft  the  wifedome,  powre  and 
Goodnes  ofGodpn,  and  by  his  Creatures ,  and  Creation  to  be  feen  andleamed. 
By  parables  and  Analogies  of  whofe  natures  and  properties,the  courfe  ofthe  Ho¬ 
ly  Scripture,  alfo,  declareth  to  vs  very  many  Myfteries.The  whole  Frame  of  Gods 
Creatures, (which  is  the  whole  world, )is  to  vs, a  bright  glafie:  from  which,  by  re¬ 
flexion,  reboundeth  to  our  knowledge  and  perceiuerance,  Beames ,  and  Radiati¬ 
on  s.-reprefenting  the  Image  of  his  Infinite  goodnes, Omnipotecy,and  wifedome. 
And  wc  therby ,  are  taughtand  perfuaded  to  Glorifie  our  Creator,as  God.-and  be 
thankefull  therfore  .  Could  the  Heatheniftes  finde  thefe  vfes,of  thefe  moft  pure, 
beawtifulLand  Mighty  Corporall  Creaturesrand  fliall  vve, after  that  thetrue  Sonne 
ofrightwifenefie  is  rifen  aboue  the  Horizon, of  our  temporall  Hemijjharieyiod.  hath 
fo  abundantly  ftreamed  into  our  hartes,the  direft:  beames  ofhis  goodnes ,  mercy, 
and  grace:  Whofe  heat  All  Creatures  feele :  Spirituall  and  Corpora)].-  Vifible  and 

b.ij.  Inui- 


lohn  D ee  his  Mathematical!  Preface » 

Inuifible.-S  hall  we (I  fay)looke  vpon  the  Heauen, Stems, and  Planets,^  an  Oxe  and 
an  AlTe  doth:  no  furder  carefull  or  inquifitiue, what  they  are.-  why  were  they  Cre¬ 
ated, How  do  they  execute  that  they  were  Created  forc'SeingJAll  Creatures,were 
for  our  lake  created  :  and  both  we,  and  they, Created,  chiefly  toglorifie  the  Al¬ 
mighty  Creator:  and  that,  by  all  meanes,to  vs  poffible.  Noliteignorare{i aith  Plate 
in  Epnomis )  Aftronomiam,  Sapientiftimu  quiddam  ejfe.  Be ye  not  ignorant jjfJlro* 

notnie  to  be  a  thyng  of  excellent  yoifedome.  cpflronomie, wzs  to  vs,from  the  be¬ 
ginning  commended, and  in  maner  commaunded  by  God  him  felfe.In  afinuch  as 
he  made  the  Sonne, CMone, and  Stems, to  be  to  vs, for  Sijgnes,an&  knowledge  ofSea- 
fons,and  for  Diftin&ions  oFDayes,and  yeares,  Many  wordes  nede  not .  But  I 
wifh,euery  man  fliould  way  this  wor d,Signes.  And  befides  that,  conferre  it  alfo 
with  the  tenth  Chapter  of  Hteremie.  And  though  Some  thinke ,  that  there, they 
haue  found  a  rod.- Yet  Modeft  Reafon,wiIl  be  indifferent  Iudge,  who  ought  to  be 
beaten  therwith,in  relped  of  our  purpofe.  Leauing  that :  I  pray  you  vnderftand 
this  :  that  without  great  diligence  of  Obferuation ,  examination  and  Calculation, 
their  periods  and  oourfes(wherby  Dift'intfion  ofSeafons,yeares,and  New  Mones 
might  precifely  be  knowne) could  not  exa&ely  be  certified  .  Which  thing  to  per¬ 
formers  that  Art ,  which  we  here  haue  Defined  to  be  Aftronomie.  Wherby ,  we 
may  haue  the  diffindl  Courfe  of  Times, dayes, yeares,  and  Ages:  afwell  for  Confi- 
deratio  of  Sacred  Prophefies,accomplifhed  in  due  time, foretold ;  as  for  high  My- 
fticall  Solemnities  holding.- And  for  all  other  humaine  affaires  ,  Conditions ,  and 
couenanres ,  vpon  certaine  time  ,betwcne  man  and  man  •  with  many  other  great 
vfcs.-  Wherin ,  ( verely)  , would  be  great  incertainty,  Confufion,vn truth,  and  bru- 
tifli  Barbaroufnes:  without  the  wonderfull  diligence  and  fkill  of  this  Arte :  conti¬ 
nually  learning, and  determining  Times, and  periodes  of  Time ,  by  the  Record  of 
the  heauenlybooke ,  wherin  all  times  are  written .-  and  to  be  read  with  an  Aftrono - 
■me  all  fidffe,  in  ftede  of  afcftue. 

Muflke  , of Motion, hath  his  Original!  caufe .-  Therfore ,  after  the  motions 

moft  fwift,and  moft  Slow, which  are  in  the  Firmament, of  Nature  perfonned:and 
vnderthe  Astronomers  Conftderation  -now  I  will  Speake  ofan  other  kinde  of Motion , 
producing  fou  nd,audible,and  of  Man  numerable.  CMufikel  call  here  that  Science, 
Which  of  the  Grekes  is  called  Harmonic* .  Not  medling  with  the  Controueriie  be- 
tweiie  the  auncient  Harmoniftes,2.rtd  Canoniftes .  Nf  ufllce  is  a  Ninth  ernaticaJJ. 

Science , which  teaebtetb,by  fenfe  and  reafon,  perfectly  to  iudge, and 
oTcfef  the  diiierfities  of  foundes,hye  and  low.  Aftronomie  and  cMuftke 
are  Sifters, faith  Plato.  As, for  Aftronomie,  the  eyes :  So,  for  Harmonious  Motion, the 
eares  were  made.  But  as  Aftronomie  hath  a  more  diuine  Contemplation ,  and  co- 
modity,then  mortall  eye  can  perceiue  :  So, is  c Muftke  to  be  confidered,that  the 
i  ,  *  Minde  may  be  preferred,before  the  care .  And  from  audible  found,  we  ought 

to  afeende ,  to  the  examination  :  which  numbers  are  Harmonious which  not. 
And  why, either,  the  fee  arc  :  of  the  other  are  not.  I  could  atiarge,in  theheauenly 
2.  *  motions  and  diftances  ,  defcribe  ameruaiIous  Harmonie  ,  o£  Pythagoras  Harpe 

^ .  with  eight  ftringes.  Aifo,fbmwhat  might  be  fay d  of  Mercurius*  two  Harpes, 
eche  of  foure  Stringes  Elementall.  And  very  ftraunge  matter, might  be  alledged 
5 .  of  the  Harmonie,  to  our  *  Spiritual!  part  appropriate.  As  in  Ptolomms  rhird  boke,  in 

the  fourth  andfixth  Chapters  may  appeare .  *  And  what  is  the  caufe  of  the  apt 
6  bondtymd  frendly  felowlhip,of  the  Intelle&uall  and  Mentall  part  of  vs ,  with  our 

groftc& corruptible  body :  but  a  certaine  Meane,  and  Harmonious  Spirituality,  with 

both 


lohn  Dee  his  Mathematical!  Preface. 


both  participatyng^  of  both  (in  a  maner)reftdtyngi  In  the *  Tune  of  Mans  <voyce,and  alp 
*  the found  of  Infrument^dmx  might  be  fay  d,  of  Harmonic:  N  o  common  Muficien  g . 

would  lightly  beleue.But  of  the  fundry  Mixture(as  I  may  terme  it)  and  concurfe,  I'D* 

diuerfe  collation, and  Application  of  th  efe  Harmonies:  as  of  thre,foure,fiue,or  mo:  Read  in  A* 
Maruailous  haue  the  efre&es  ben:  and  yet  may  be  foundc,and  produced  the  like:  r^°dls!ns  - 
with  feme  proportional!  confidcration  for  our  time, and  being  :  inrdpedt  of  the 
State ,  of  the  thinges  then :  in  which ,  and  by  which ,  the  wondrous  effe&es  were  tH^'an£ 
wrought.  Democritus  and  T heophrasius  affirmed,  that, by  CELufike,  griefes  and  di-  y. chapters, 
feafes  of  the  Minde,and  body  might  be  cured, or  inferred.  And  we  finde  in  Re-  where you 
cord e, that  T er pander,  Anonffmemas^  Orpheus  ^Amphion,Dauid,  Pythagoras^  Empedo-  Jkall  bane 
cles,  jjclepiadesymd  T imotheus, by  Harmonicall  Confonacy,haue  done, and  brought  fom£  occafion 
to  pas, thinges, more  then  meruailous ,  to  here  of.  Of  them  then,  making  no  far-  ffder  to 
der  difcourfe,in  this  place  :  Sure  I  am,  that  Common  Mufike ,  commonly  vfed,  is 
found  to  the  c Muficiens  and  Hearers,to  be  fo  Commodious  and  pleafant ,  That  if  commonly  is 
I  would  fay  and  difpute,but  thus  much:  That  it  were  to  be  otherwife  vfed ,  then  it  thought. 
is, I  iliould  finde  more  repreeuers,  then  I  could  finde  priuy,or  fkilfull  of  my  mea¬ 
ning.  In  thinges  therfore  euident,and  better  knowen,then  I  can  exprefTe:andfo 
allowed  and  liked  of,  (as  I  would  wilh/ome  other  thinges, had  the  like  hap)  I  will 
{pare  to  enlarge  my  lines  any  farder,but  confequently  follow  my  purpofe. 

Of  Cofmograpllie,!  appointed  briefly  in  this  place,  to  gcue  youforae 


intelligence.  Cofmographie,is  the  whole  and  perfed  defeription  of 
the  heauenly,and  alfo  elementall  parte  of  the  world  ,  and  their  ho* 
mologall  application ,  and  mutuall  collation  necelfarie.  This  Art, 
requireth  AHronomie ,  Geographic ,  Hydrographie and  CMttfike .  Therfore ,it  is  no 
finall  Arte, nor  fo  fimple,as  in  common  pra&ife,  itis(iiightly)confidered.  This 
marcheth  Heauen,  and  the  Earth,in  one  frame,and  aptly  applieth  parts  Correfpo- 
dentrSo  ,as,  the  Heauenly  Globe,  may  (in  pradife)  be  ducly  deferibed  vpon  die 
Geographicall ,  and  Hydrographical!  Globe .  And  there ,  for  vs  to  confider  an 
Aqutnociiall  Circle ,  an  Ecliptike  line ,  Colures,  Poles,  Stems  in  thei  r  true  Longitudes, 
Latitudes,Declinations,and  Verricajitie.-alfb  Climes, and  Parallels:and  by  an  Ho¬ 
rizon  annexed, and  reuolution  of  the  earthly  Globe(as  the  Heauen,is,  by  the  Pri- 
fnduantfi  tied  about  in  24.requall  Houres)  to  learne  the  Rifinges  and  Settinges  of 
S  terres  (of Virgill in  his  Georgikes:  of  Hefod:  of Hippocrates  in  his  Medicinall Sphere,  to 
Perdicca  King  of  the  Macedonians:  of  Diodes  pto  King  Antigonus ,  and  of  other  fa¬ 
mous  P  hilo fop  hers  prefcriBecra  thing  necefTary,for  due  manuring  of  the  earth ,  for 
Nauigation, for  the  Alteration  ofmans  body:being,whole,Sicke,wounded,or  bru- 
fed.  By  the  Reuolution ,  alfo ,  or  mouing  of  the  Globe  Cofmographicall ,  the 
Riling  and  Setting  of  the  Sonne:  thcLengthcs,ofdaycs  and  nightes :  the  Houres 
and  times  (both  night  and  dayjareknowne :  with  very  many  other  pleafant  and 
neceffaty  vfes  :  Wherof,  fome  are  known  chut  better  remaine, for  fuch  to  know 
and  vfe.  who  of  a  fparke  of  true  fire,can  make  a  wonderfull  bonfire,  by  applying  of  *■>« 
due  matter, duely.  1  rr 3  ° 

Of  Aftrologie ,  here  I  make  an  Arte,  feuerall  from  AHronomie :  not 
by  new  deuife,  but  by  good  reafon  and  authoritie :  for,  Aftrologie, is  an  Arte 
Mathematical! ,  which  reafonably  demonftrateth  the  operations 
and  effedles,  of  the  naturall  beames,  of  light,  and  fecrete  influence: 
of  the  Sterres  and  P  lanets. :  in  euery  element  and  elementall  body: 

b.iih  at 


lohn  Dee  Lis  Mathem  aticall  P  raeface , 

at  all  times  ,  in  any  Horizon  affigned .  This  Arte  is  furniihed  with  ma¬ 
ny  other  great  Artes  and  experiences:  As  with  perfe&e  Perftecliue,  ^JHronomic, 
Cofmographie,  Natural!  Philofophie  of  the  /{..Elementes, the  Arte  of  Graduation, and 
fome  good  vnderftading  in  CMuftke :  and  yet  moreouer,  with  an  other  great  Arte, 
hereafter  following,  though  I,  here,  fet  this  before,  for  fome  confiderations  me’ 
mouing.  Sufficient  (you  fee)  is  the  ftuffe,  to  make  this  rare  and  fecrete  Arte, of: 
and  hard  enough  to  frame  to  the  Conclufion  Syllogifticall  .  Yet  both  the  mani- 
fblde  and  continuall  trauailes  of.  the  raoft  auncient  and  wife  Philofophers,for  the 
atteyning  of  this  Arte :  and  by  examples  of  effedes ,  to  confirme  the  fame ;  hath 
left  vnto  vs  fufficient  proufe  and  witndfe  •  and  we,alfo,daily  may  pcrceaue ,  That 
mans  body,  and  all  other  Elementall  bodies,  are  altered,  difpofed,  ordred,  pleafu- 
red,  and  difpleafured,  by  the  Influential!  working  of  the  Sunne,Monc, and  the  other 
Starres  and  Planets  .  And  therfore,fayth AriBotle^  in  the  firft  of  his  Meteorological!. 
bookes,  in  thefecond  Chapter Eft  antem  necejfario  Mundus  iUefitpernis  lationibus 
fere  contimus ;  V t,  mde,  vis  tins  vniuerfta  regatur .  Ea  ftquidcm  Caufa  puma  pit  and  a 
omnibus  eft,  vndemotus  principium  exisnt.  That  is:  This  i  Elementally  World  is  of 
necefiitie,  almojl ,  next  adioyning/o  the  heauenly  motions :  T  hat  from  thence , 
nil  his  loertne  or  force  may  hegouerned.  For /hat  is  to  he  thought  thefirf  Caufe 
'Pntoall ;  from  y>hich,the  beginning  of  motion, is .  And  againe,  in  the  tenth 
Chapter,  Opcrtet  igitur  &  bomm  principia  fumamns ,  (ft  cattfits  omnium  ftmiliter. 
Principium  igitur  vt  mouenspracipuumcf  (ft  omnium  primum ,  Cir cuius  tile  eftjn  quo 
manifefte  S oils  latio ,  &c .  Andfo  forth .  His  Meteor ologicall  bookes,  are  full  ofargu- 
mentes,and  effeduall  demonftrations,ofthe  vertue,  operation,  and  power  of  the 
heauenly  bodies,  in  and  vpon  the  fower  Elementes,  and  other  bodies, of  them 
(either  perfectly, or  vnperfedly  )  compofed,  Arid  in  his  fecond  booke.  Be  Genera- 
tione  drCorruptione ,  in  the  tenth  Chapter ,  flup  circa  eftprima  latio ,  Ortusfr  Interi- 
tus  caufa  non  eft:  S ed  obliqui  Circuit  latio :  ea  namift  (ft  continua  ejlftft  daobus  motibus  ft: 
In  Erigliffie,  thus .  Wherefore  the  uppermost  motion,  is  not  the  caufe  of  Gene * 
ration  and  Corruption,  but  the  motion  of  the  Zodiake :  for ,  that ,  both,  is  con * 
tinuall ,  and  is  caufed  of  two  mouinges .  And  in  his  fecond  booke,  and  fecond 
Chapter  of  hys  Phyfikes.  Homo  nam^generat  hominem,  at %  Sol.  For  Man  (lay  th  he) 

and  the  Sonne,  are  caufe  of  mans  generation .  Authorities  may  be  brought, 
very  many :  both  of  1000.2000.yea  and  3000.  y.eares  Antiquitie :  of  great  Philo- 
fophers.  Expert,  Wife,  arid  godly  men,for  that  Conclufion:  which,daily  and  houre- 
dy,we  men,may  difeerne  and  perceaue  by  fenfe  andreafon  :  All  beaftes  do  feele, 
andfimplyihew,  by  their  actions  and  paffions,  outward  and  inward  ;  All  Plants, 
Herbes,  Trees,  Flowers,  and  Fruites .  And  finally,  the  EIementes,and  all  thinges 
of  the  Elementes  compofed,  do  geue  Teftimonie  (as  Ariftotle  fayd)  thattheyr 
Whole  Fdijpoftions,  yertues ,  andnaturall  motions,  depend  of  the  jiffiiuitie  of 
tl heauenly motions  and  Influences .  Whereby,  hefide  the  jftecifcaU  order  and 
forme, due  to  cuery  feede:  and  hefide  the  Flature ,propre  to  the  Indiuiduall ' Ma • 
trix,  of the  thing  produced:  What fhall  he  the  heauenly  Imprefiion,the  perfeCf 
and  circumfieffe  JFlrologien  hath  to  Conclude.  N  ot  onely  (by  Apotelefmesft 0  orb 
but  by  Nat-urall  and  Mathematical!  demonftratipn  Whereunto,  what 

Sciences  are  reqtiifitc  (  without  exception  )  I  partly  haue  here  warned:  And  In  my 
Propadmmes  (  befides  other  matter  there  dficlofed  )  I  haue  Mathematically  furni- 
ihed  vp  the  whole  Method ;  To  this  our  age, not  fo  carefully  handled  by  any,that 

euer 


lohn  Dee  his  Mathematical!  Preface. 

cuer  I  faw, or  heard  of.  I  was,  (for  *  2i.yeares  ago)  by  certaine  earneft  difpUtati-  *  Anno.  t  54S 
ons,of  the  Learned  Gerardus  Mercator ,and  Antomus Oogaua,  (and  other, )  therto  fo  and  1 540. ?;j 
prouoked:and(by  my  conftantand  inuirtcible  zeale  to  the  veritie)in  oblcruations  Louayn, 
ofHeauenly  Infliienciesf  to  the  Minute  of  time,  )than,fo  diligent:  And  chiefly  by 
the  Supernatural!  influence,frorn  the  StarreofIacob,(b  directed  .-That  any  Modeft 
and  Sober  Student, carefully  and  diligently  fekingforthe  Truth,  will  both  finde 
&  cofefie,  therin,to  be  the  Veritie, of  thefe  my  wordes:  And  alfo  become  a  Reafo- 
nable  Reformer,  of  three  Sortes  of people:  about  thefe  Influentiall  Operations, 
greatly  erring  from  the  truth.  Wherof,  the  one ,  is  Light  Beleuers,the  other,  Note* 

Light  Del|)ilers,and  the  third  Light  Pracflifers.  The  firft,& moft  comon 

Sort,  thinke  the  Heauen  and  Sterres, to  be  anfwerable  to  any  their  doutes  or  de-  *  * 
fires:  which  is  not  fo;and,in  dedc,they,to  much,ouer  reache.  The  Second  forte 
thinke  no  Influentiall  vertue  (  fro  the  heauenly  bodies  )  to  beare  any  Sway  in  Ge-  *  * 

neration  and  Corruption, in  this  Hlementall  world.  And  to  the  Sunne  ,  Mone  and 
Sterres  ( beingfo  many,fo  pure,fo  bright ,  fo  wonderfull  bigge ,  fo  farre  in  diftancc, 
fb  manifold  in  their  motions ,  fo  conftant  in  their  periodes .  &c  . )  they  affigne  a 
fleight, Ample  office  or  two,and  fo  allow  vnto  the(according  to  their  capacities)as 
much  vertue, and  power  Influentiall,as  to  the  Signe  of  the  Sunne,  Mone,  and  feuen 
Sterres,  hanged  vp(for  Signesjin  Lon  don, for  diftindion  ofhoufes ,  &  fuch  groffe 
helpes,in  our  wordly  affaires:  And  they  vnderftand  not(or  will  not  vnderftand)  of 
the  other  workinges,and  vertues  of  the  Heauenly  Stinne,Mone,  and  Sterres  .•  notfo 
much, as  the  Mariner,or  Hufband  man  :  no ,  not  fo  much, as  the  Elephant  doth,  as 
the  Cynocephalus ,  as  the  Porpentinc  doth :  nor  will  allow  thefe  perfed ,  and  incor¬ 
ruptible  mighty  bodies,  fo  much  v-ertuall  Radiation,  &  Force ,  as  they  fee  in  a  litle 
peece  of  a  Magnes  Hone: which,  at  great  diftance,fheweth  his  operation .  And  per- 
chaunce  they  thinke, the  Sea  &  Riuers  (  as  the  Thames )  to  be  fome  quicke  thing, 
and  fo  to  ebbe,and  flow,  run  in  and  out,  of  them  felues,at  their  owne  fantafies. 

God  helpe,God  helpe.  S urely, thefe  men,come  to  fhort :  and  either  are  to  dull; 
or  willfully  blind:  or, perhaps, to  malicious  .  The  third  man,  is  the  common  and 
vulgare  '^Aftrologien,ot  Pradifer :  who,  being  not  duely,artificially,and  perfedly  3 . 
furnifhed:yet,either  for  vaine  glory,or  gayne :  or  like  a  Ample  dolt,  &  blinde  Bay¬ 
ard,  both  in  matter  and  maner,erreth:to  the  difcredit  of  the  Wary ,  and  modeft  A- 
J?roJoricn:cLnd  to  the  robbing  of  thofe  moft  noble  corporall  Creatures,of  their  Na¬ 
tural!  Vertue:  being  moft  mighty :  moft  beneficiall  to  all  elementall  Generation, 
Corruptiorfand  the  appartenances  -  and  moft  Harmonious  in  their  Monarchic: 

For:  which  thinges, being  knowen, and  modeftly  vfed:we  might  highly,and  conti¬ 
nually  glorifie  God, with  the  princely  Prophet,  faying.  J  he  Heauens  declare 

the  Glorie  of  GodtSoho  made  the  Heaues  in  his  wife  dome:  "Soho  made  the  Sonne , 
for  tohaue  dominion  of the  day :  the  Mone  and  Sterres  to  hauc  dominion  of the 
nygbt:  whereby,®  ay  today  loiter eth  talke:  and  night, to  night  dedaretb  know* 
ledge.tprayje  him, ally e  Sterres, and  Light.  Amen. 

JN  order,  nowfoloweth ,  of Stfttlkc,fomewhat  to  fay,  what  we  meane  by 
that  name: and  what  commodity, doth, on  fuch  Art,  depend.  Statike ,  is  an 
Arte  Mathematicall, which  demonflrateth  the  caufes  ofheauynes, 
and  lightnes  of  all  thynges :  and  of  motions  and  properties ,  to  hea- 
uynes  and  lightnes  ,belonging.  And  for  afmuch  as,  by  the  Bilanx ,  or  Ba¬ 
lancers  the  chieffenfible  Inftrument , )  Experience  of  thefe  demonftrations  may 

b.iiij.  be 


lohn  Dee  his  Mathematical!  Preface. 

be  had:  we  call  this  Anfpatike:dn2it  isjhe  Experimentes  of  the  Balance.  Oh,  that  men 
wift,  what  proffit,(aIl  maner  of  wayes)by  this  Arte  might  grow,  to  the  hable  exa- 
»  miner,and  diligent  pra&ifer.  Thou  onely,knoweft all  thinges  precifely  (O  God) 
■”  who  haft  made  weight  and  Balance, thy  Iudgement:.  who  haft  created  all  thinges 
in  Number ^Waight,  and  haft  way  ed  themountaines  andhilsina  Ba- 

lance:  who  haft  peyfed  in  thy  hand ,  both  Heauen  and  earth .  We  therfore  war¬ 
's  ned  by  the  Sacred  word, to  Confider  thy  Creatures.-and  by  that  conftderation,  to 
”  Wynne  a  glyms{asitwere,  )or  lhaddowof  perceiuerance  that  thy  wiledome, 
>>  might,  and  goodnes  is  infinite,and  vnfpeakable,in  thy  Creatures  declared :  And 
”  being  farder  adder  tiled ,  by  thy  mercifull  goodnes ,  that  ,three  principal!  wayes, 
••  were,ofthe,vfed  in  Creation  ofall  thy  Creatures ,  namely ,  Number^  Waight  and 
'»*  OHeafure, And  for  as  much  as,of  2N lumber  and  Meafure, the  two  Artes(aundent,  fa¬ 
s’  mous,and  to  humaine  vfes  moft  neceflary, )  are, ail  ready, fufficiently  kno'wen'  and 
js  extant:  This  third  key ,  we  befeche  thee  ( through  thy  accuftomed  goodnes,) 
«  that  it  may  come  to  thenedefull  and  fufficientknowledge,offuch  thy  Seruauntes, 
”  as  in  thy  workemanlhip ,  would  gladly  finde, thy  true  occaftons  (purpofely  of  the 
>»  vfed )  whereby  we  fhould  glorifie  thy  name, and  ftiew  forth  (to  the  weaklinges  in 
faith)  thy  wondrous  wifedome  and  Goodnes.  Amen. 

Meruaile  nothing  at  this  pang(godly  frend,you  Geiidc  and  zelous  Student.) 
An  other  day, perchaunce, you  will  perceiue,  what  occafion  moued  me.  Here,  as 
now, I  will  giue  you  fome  ground ,  and  withallfomelhew ,  of  certaine  commodi¬ 
ties, by  this  Arte  arifing.  And  bycaufe  this  Arte  is  rare ,  my  Wordes  and  pradlifes 
might  be  to  darke :  vnleaftyou  had  fome  light, hoiden  before  the  mattenand  that, 
beft  will  be, in  giuing  you,out  of  Archimedes  demonftrations,a  few  principal  Con- 
cluftons,as  folowerh. 

1. 

The  Superficies ofeuery  Liquor,  byitfelfe  confiflyng ,and  in 
quyet,  is  Sphxricall  :  the  centre  whereof,  is  the  fame ,  which  is  the 
centre  of  the  Earth. 

2.  .  , 

If  Solide  Magnitudes, being  of  the  fame  bignes,  or  quatitie,  that 
any  Liquor  island  hauyngalfo  the  fame  Waight :  be  let  downe  in¬ 
to  the  lame  Liquor, they  will  fettle  downeward,fo,that  no  parte  of 
them,fhall  be  aboue  the  Superficies  of  the  Liquor  :  and  yet  neuer- 
theles ,they  will  not  finke  vtterly  downe,or  drowne. 

V 

If  any  Solide  Magnitude  beyng  Lighter  then  a  Liquor ,  be,  let 
downe  into  the  fame  Liquor ,  it  will  fettle  downe,  fo  farre  into  the 
fame  Liquor,  that  fo  great  a  quantitie  of  that  Liquor,  as  is  the  parte 
of  the  Solid  Magnitude,  fettled  downe  into  the  fame  Liquor :  is  in 
Waight, squall, to  the  waight  of  the  whole  Solid  Magnitude. 

4* 

Any  Solide  Magnitude ,  Lighter  then  a  Liquor ,  forced  downe 

into 


lohn  Dee  his  Mathematical!  Preface. 

ifjto  the  fame  Liquor ,  will  moue  vpward  ,  with  fo  great  a  poweiq 
by  how  much ,  the  Liquor  hauyng  xquall  quantitie  to  the  whole 
Magnitude's  heauyer  then  the  fame  Magnitude, 


J- 


/ 


Any  Solid  Magnitude, heauyer  then  a  Liquor, beyng  let  do  wne 
into  the  fame  Liquor, will  linke  downe  vtterly :  And  wilbe  in  that 
Liquor  ,  Lighter  by  fo  much ,  as  is  the  waight  or  heauynes  of  the 
Liquor, hauing  bvgnes  or  quantitie, squall  to  the  Solid  Magnitude, 


6, 


i.V. 

If  any  Solide  Magnitude ,  Lighter  then  a  Liquor  ,  be  let  downe  Sphar  e  according  to 

1  J  ^  j  i  r  r*  ^  |  |  -111  any  proportion  af- 

into  theiame  Liquor ,  the  waight  or  the  lame  Magnitudewill  be, 
to  the  Waight  of  the  Liquor . (Which  is  aquallin  quantitie  to  the 
whole  Magnitude,)  in  that  proportion ,  that  the  parte ,  of  the  Mag- 
nitude  fettled  downe, is  to  the  whole  Magnitude. 

gY  thefe  verities  ,  great  Errors  may  be  reformed, in  Opinion  of  the  Natural! 

Motion  of  thinges, Light  and  Heauy.  Which  errors, are  in  Natural!  Philofophie 
(almoft  )  of  all  me  allowed.-to  much  trufting  to  Authority /and  falfe  Suppofitions. 

As, Of  any  two  body es, the  heauyer,  to  moue  downward  falter  then 
the  lighter.  This  error,is  notfirftby  me, Noted;  but  by  one  lohn  Baptist  de 
nedichs.  Thechief  of  his  propofitions,is this:  which  feemeth  a  Paradox. 

If  there  be  two  bodyes  of  one  forme,  and  of  one  kynde,  asquail  in  . 

quantitie  or  vnasquall ,  they  will  moue  by  aequall  fp ace,  in  xquall  A*ar  °X* 
tymeiSo  that  both  theyrmouynges  be  in  ay  re ,  or  both  in  water ;  or 
in  any  one  Middle. 

Hereupon ,  in  the  feate  of  Gunny  ng,certar  e  good  difeourfes  (  otherwife)  jV.  T. 
may  receiiie  great  amen  dement,  and  furderance.  In  the  emended  purpofe ,  alio,  rj-i  .  < 

allowing  fomwhat  to  the  imperfection  df Nature :  not  aunfwerable  to  the  preci- 
fenes  oldemon;f  ration .  Moreouer,by  the  forefaid  propofitiofts  (  wifely  vied.)  thefe  Prof  oft - 
The  Ayre,the  water, the  Earth, the  Fire,  may  be  nerely,knowen,how  light  or  hea-  t ions. 
uy  th  y  arc  { N  aturally )  in  their  affigned  partes  :  or  in  the  whole.  And  then,to 
thinges  Eiementall,turningvourprad:ife:  you  may  deale  for  the  proportion  of the 
Eiementes ,  in  the  thinges  Compounded .  Then,  to  the  proportions  of  the  Hu¬ 
mours  in  Man:  their  waightes:  and  the  waight  of  his  bones,  and  fiefh.&c.  Than, 
by  waight, to  haue  confideration  of  the  Force  of  man, any  maner  ofwav:  in  whole 
or  in  pai  t.Then  ,may  you,  ofShips  water  drawing ,  diucrfly,in  the  Sea  and  in  frefh 
water,  haue  plealant  confideration and  ofwaying  vp  of  any  thing,  fonken  in  Sea 
or  mftefli  water  &c .  And  (to  lift  vp  your  head  a  loft :  )  by  waight, you  may, as 
precifely,as  by  any  inftrument  els,meafure  the  Diameters  of  Sonne  and  eMone.&c. 
Frendejpray  you ,  way  thefe  thinges, with  the  iuft  Balance  ofReafon.  And  you 
will  hnde  Memailes  vpon  Meruailes  ;  And  gfteme  one  Drop  of  Truth  (  yea  in 
Naturall  Phtlo,opnie)more  worth,  then  whole  Libraries  of  Opinions,  vndemon- 
ftrated:  or  not  aunfwenng  to  Natures  Law,and  your  experience.  Leaning  thefe 

c-i*  thinges. 


the  waight  of  the 
S phare  thereto 
£wymming> 


A  common 
error  jioted* 


T he  practije 
Statical l,  to 
kriovp  the  pro¬ 
portion,  be- 
txvene  the 
Cube,  and  the 
Sphare 


J.D, 

*  For,  Jo,  bane 
you.  2  5  6, 

partes  of  a 
Gratae. 


*The  proportion  of 
the  Square  to  the 
Circle  infiribed. 


*The  Squaring, of 
the  Circle,Mecha~ 
nically . 

*To  any  Square 
geuen,  to  gene  a 
Circle,  equaU. 


lohn  Dee  his  Mathematical!  Preface . 

thinges,thus:I  will  giue  you  two  or  three,light  pra&ifes,  to  great  purpofe :  and  fb 
iinifli  my  Annotation  StaticalL  In  Mathematical!  matters ,  by  the  Mechanidens 
ayde ,  we  will  behold,  here,  the  Commodity  of waight .  Make  a  Cube,of  any 
one  Vniforme :  and  through  like  heauy  ftufFe:  of  the  lame  Stuffe,make  a  Sphere 
or  Globe,precifdy,of  a  Diameter  squall  to  the  Radicall  hdc  of  the  Cube .  Your 
ftuffe,may  be  wood,  Copper,  Tinne,  Lead,SiIuer.&c.  (being, as  I  fayd,oflike  na¬ 
ture  ,  condition, and  like  waight  throughout. )  And  you  may ,  by  Say  Balance, 
haue  prepared  a  great  number  of  the  Imalleft  waightes  :  which,  by  thofe  Balance 
can  be  difeerned  or  tryed.-and  fo,haue  proceded  to  make  you  a  perfed  Pyle,  com¬ 
pany  &  Number  of  waightes:  to  the  waight  oflix,eight,or  twclue  pound  waight: 
moft  diligently  tryed,all.And  of  euery  one  ,  the  Contentknowen,  in  your  lead 
waight,that  is  wayable.  [They  that  can  not  haue  thele  waightes  of  precifenes : 
may, by  Sand, Vniforme, and  well  dufted,makc  them  a  number  of waightes, fome- 
whatnerepredfenes  :  by  hailing  euer  the  Sand :  they  lliall  ,at  length,  come  to  a 
•lead  common  waight.Therein,I  leaue  the  farder  matter, to  their  difcretion,wiiom 
nedelliali  pinche.]  Th  e.  Venetians  conftderation  of  waight ,  may  feme  prccife 
enough:by  eight  delcentes  progrefsionall,*  hailing ,  from  a  grayne.  Your  Cube, 
Sphere, apt  Balance, and  conuenient  waightes,being  ready -fall  to  worker .  Firft, 
way  your  Cube.Note  the  Number  of  the  waight .  Wav, after  that,  your  Sphere. 
N  ore  likewife,th  e  N  uber  of  the  waight.If yo  u  now  find  the  waight  ofyour  C  ube, 
to  be  to  the  waight  of  the  Sph^re,as  21 .  is  to  n.-Then  you  fee,  how  the  Mechani- 
cien  and  Experimenter ,  without  Geometrie  and  Demonftration,  are  (  as  nerely  in 
effed)£ought  the  proportion  of  the  Cube  to  the  Sphere  :  as  I  haue  demonftrated 
it,in  the  end  of  the  twelfth  boke  of  Euclide.  Often ,  try  with  the  lame  Cube  and 
SphcCre.Then,chaunge,your  Sphere  and  Cube,to  an  other  matter:  or  to  an  other 
bignes  :  till  you  haue  made  a  perfed  vniuerfall  Experience  ofit.  Pofsible  it  is, 
that  you  lliall  wynne  to  nerer  termes,in  the  proportion. 

When  you  haue  found  this  one  certaine  Drop  of  Naturall  veritic,procede  on, 
to  Inferre,and  duely  to  make  allay, of  matter  depending.  As,  bycaule  it  is  well  de¬ 
monftrated  ,  that  a  Cylinder ,  whofe  heith ,  and  Diameter  of  his  bafe,is  squall  to 
the  Diameter  of the  Sphere  ,is  Sefquialterto  the  fame  Sphere  (thatis,as  3.  to  2:) 
To  the  n  umber  of  the  waight  of  the  Sphere, adde  halfe  fo  much, as  it  is  :  and  fo 
haue  you  the  number  of  the  waight  of  that  Cylinder.  Which  is  alfo  Compre¬ 
hended  of  our  former  Cube:  So,that  the  bafe  of  that  Cylinder ,  is  a  Circle  deferi- 
bed  in  the  Square ,  wrhich  is  the  bafe  of  our  Cube.  But  the  Cube  and  the  Cy- 
linder, being  both  of  one  heith ,  haue  their  Bafes  in  the  fame  proportion  ,  in  the 
which, they  are ,  one  to  an  other,  in  their  Mafsines  or  Soliditie .  But,before,we 
haue  two  numbers, exprelsing  their  Mafsines  ,  Solidities  ,  and  Quantities ,  by 
Waight  :wherfo  re,  we  haue  *  the  proportion  of  the  Square,to  the  Circle,  inlcribed 
in  the  fame  Square.  And  fo  are  we  fallen  into  the  knowledge  fenlible,  and  Expe¬ 
rimental!  of  ^Archimedes  great  Secret:  of  him ,  by  great  trauaile  of  minde ,  fought 
and  found.  WherFore,to  any  Circle  giuen ,  you  can  giue  a  Square  squall :  *  as 
I  haue  taught, in  my  Annotation, vpon  the  lirft  propolition  of  the  twelfth  boke. 
And  likewile,to  any  Square  giuen, you  may  giue  a  Circle  squall:  *Ifyou  deferibe 
a  Circle,which  lliall  be  in  that  proportion,  to  your  Circle  inlcribed,  as  the  Square 
is  to  the  fame  Circle  -This,you  may  do,by  my  Annotations, vpon  the  lecond  pro¬ 
polition  of  the  twelfth  boke  of  Euclide ,  in  my  third  Probleme  there.  Your  dili¬ 
gence  may  come  to  a  proportion, of  the  Square  to  the  Circle  inlcribed ,  nerer  the 
tmth,then  is  the  proportion  of  14.10  n.  And  confider,  that  you  may  begyn  at 
the  Circle  and  Square,  and  fo  come  to  conclude  of  the  Sph^re,&  the  Cube, what 


John  Dee  his  MathematicathBr.xface. 

their  proportion  is:as  now ,  yon  came  from  the  Sphere  to  the  Circle.  For, of  Sib 
uer,or  Gold, or  Latton  Lamyns  or  plates  (thorough  one  hole  drawees  the  manef 
is)if  you  make  a  Square  figure.-&  way  itrand  then,defcrib.ing  theron,  the  Circle  in 
fcribed:&  cut  of,&  file  away,precifely  (to  the  Circle)  the  ouerplus  of  the  Square: 
you  lhall  then,waying  your  Circle ,  fee, whether  the  waight  of  the  Square ,  be  to 
your  Circle,  as  14.  ton.  As  I  haue  Noted ,  in  the  beginning  of  Cuchdes  twelfth 
boke.&c.after  this  refort  to  my  laft  propofition,vpon  the  laft  of  the  twelfth .  An  d 
there, helpe  your  felfe, to  the  end.  And,  here,  Note  this,  by  the -way .  That  we 
may  Square  the  Circle ,  without  hauing  knowledge  of  the  proportion, of  the  Cir¬ 
cumference  to  the  Diameter :  as  you  haue  here  perceiued .  And  otherwayes 
allb,  I  can  demonftrate  it.So  that, many  haue  cumberd  them  felues  fuperfluoiifly, 
by  trauailing  in  that  point  firft ,  which  was  not  of  neeefsitie,fijrft :  and  alfb  very  in¬ 
tricate.  And  eafily,you  may,  (and  that  diuerfly)  come  to  the  knowledge  of  the 
Circumference:  the  Circles  Quantitie ,  being  firft  knowen.  Which  thing,!  leaue 
to  your  confideratiommaking  haft  to  defpatch  an  other  Magiftrall  Probleme:  and 
to  bring  it,nerer  to  your  knowledge,and  readier  dealing  with, then  the  world  (be¬ 
fore  this  day,) had  it  for  you, that  I  can  tell  of. And  that  is,  A  Mechmicall  Dubblyng 
of  the  Cube:&c.  Which  may,  thus, be  done:  Make  of  Copper  plates, orTyn 
plates, a  fourfquare  vpright  Pyramis,or  a  Cone:  perfectly  fafhioned 
in  the  holow, within  .  Wlierin,  let  great  diligence  be  yfed ,  to  ap~ 
proche  (as  nere  as  may  be )  to  tbe  Mathematical!  perfection  of  thofe 
figures .  At  their  bafes, let  them  be  all  open :  euery  where,  els,  moll 
dole, and  iufl  to.  From  the  vertex,  to  the  Circumference  of  the  bafe 
of  the  Cone:  &  to  the  fides  of  the  bafe  of  the  Pyramis  :Let  q.flraight 
lines  be  drawen,in  the  infide  of  the  Cone  and  Pyramis :  makyng  at 
their  fall, on  the  perimeters  of  the  bafes ,  equall  angles  on  both  fides 
them  felues  ,  with  the  fayd  perimeters .  Thefe  4. lines  ( in  the  Pyra¬ 
mis  :andas  many, in theCone)diuide:one,in  12.  ^quallpartes  :  and 
an  other, in  24. an  other, in  60 ,  and  an  other,  in  100  .  (reckenyng  vp 
from  the  vertex. )  Or  vfe  other  numbers  of  diuifion ,  as  experience 
fh all  teach  you.  Then,*  fet  your  Cone  or  Pyramis,  with  the  vertex 
downward ,  perpendicularly ,  in  refped  of  the  Bafe.  (Though  it  be 
otherwayes, it  hindreth  nothyng.)  So  let  the  moll  fledily  be  flayed. 
N  ow, if  there  be  a  Cube,  which  youwold  haue  Dubbled.Make  you  a  prety  Cube 
of  Copper,  Siluer,  Lead,  Tynne,  Wood,  Stone,  or  Bone.  Or  els  make  a  hollow 
Cube, or  Cubik  coffen,  of  Copper,  Siluer,  Tynne,or  Wood  &c .  Thefe, you  may 
fo  proportio  in  refpeft  ofyour  Pyramis  or  Cone  ,  that  the  Pyramis  or  Cone,  will 
be  liable' to  conteine  the  waight  of  them,  in  water,  3  .or  4.  times  :at  the  leaft:  what 
ftufffo  euer  they  be  made  of.Let  not  your Solid  angle ,  at  the  vertex,be  to  lharpe: 
but  that  the  water  may  come  with  eafe,to  the  very  vertex,of  your  hollow  Cone  or 
Pyramis.Put  one  ofyour  Solid  Cubes  in  a  Balance  apt:  take  the  waight  therof  ex¬ 
actly  in  water  .  Powre  that  water,  (  without  Ioffe  )  into  the  hollow  Pyramis  or 
Cone>quietly.  Marke  in  your  lines, what  numbers  the  water  Cutteth :  Take  the 
waight  of  the  lame  Cube  again  e  .•  in  the  lame  kinde  of  water ,  which  you  had  be¬ 
fore  :  put  that  *  alfo,  into  the  Pyramis  or  Cone, where  you  did  put  the  firft.  Marke 
now  againe,  in  what  number  or  place  of  the  lines,  the  water  Cutteth  them.  Two 

c.ij.  wayes 


Note 

Squaring  of 
the  Circle 
"yitbotuknowt 
ledos  of  the 
proportion  ve- 
tveene  Cir¬ 
cumference 
and'Diame- 
ter. 


ToOubble 

the  Cube  re - 
dilj :  by  Art  - 
Afechanicall : 
depending  vp-  1 
pen  ‘Demon-  - 
ilration  Add- 
thematic  all. 


1.  D. 

T he  ^..fidos  of  this 
Tyramis  muft  be  4. 
Ifofceies  Triangles  4 
like  mdaqmU. 


T.D. 

* In  all  workinget 
with  this  Pyramis 
or  Cone, Let  their 
Situations  be  in  all 
Pointes  and  Condi¬ 
tions,  a  like, or  all 
anetwhileyouare 
about  one  worke.Elf 
you  will  ene. 


I.D. 

*  Conjider  well  whan 
you  muft  put  your 
waters  togyther:  and 
whan, you  muft  emp¬ 
ty  your first  water, 
out  of  your  Pyramis 
or  Cone.  Els  yeti 
WiU  erre. 


*  Vlitruuius. 
Lib.p.Cap.y, 

GT 

God  be  than¬ 
ked  for  this 
Jnuention,dr 
thefrnite  en- 

-*Note. 


Note,  .as  con¬ 
cerning  the 
Spharicall 
Superficies  of 
the  Water. 


KF  » 

>? 

?? 

Ji 

JJ 


*Note. 


Note  this  A- 
bridgement  of 
Dabbling  the 
pube.&C' 


Iohn  D  ee  his  Mathematical!  Preface. 

wayes  you  may  conclude  your  piirpofe :  it  is  to  wete ,  either  by  numbers  or  lines. 
By  numbers  :  as, if  you  diuide  the  fide  of  your  Fundamental!  Cube  into  fo 
many  squall  partes,  as  it  is  capable  of,conueniently,with  your  cafe ,  and  pre- 
cifenes  of  the  diuifion  .  For,  as  the  number  of  your  firft  and  leffe  line  ( in  your 
hollow  Pyramis  or  Cone,)  is  to  the  fecond  or  greater  (  both  being  counted 
from  the  vertex)  fofhall  the  number  of  the  fide  of  your  Fundamentall  Cube, 
be  to  the  nuber  belonging  to  the  Radicall  fide, of  the  Cube, dubble  to  your  Fun- 
dam entail  Cube:  Which  being  multiplied  Cubik  wife, will  fone  fhew  it  felfe,  whe¬ 
ther  it  be  dubble  or  no ,  to  the  Cubik  number  ofyour  Fundamentall  Cube .  By 
lines, thus;  As  yoiir  lefleand  firft  line, (in  your  hollow  Pyramis  or  Cone,)is  to  the 
fecond  or  greater, fo  let  the  Radicalfide  ofyour  Fundametall  Cube, be  to  a  fourth 
proportionall  line ,  by  the  1 2 .  propofition,  of  the  fixth  boke  of  Euclide  .  Which 
fourth  line,fhall  be  the  Rote  Cubilqor  Radicallfide  of  the  Cube ,  dubble  to  your 
Fundamentall  Cube :  which  is  the  thing  we  defired .  For  this, may  I  ( with  ioy) 
lay, eyphka,  eyphka, eyphka :  thanking  the  holy  and  glorious  Trinity:  hauing 
greater  caufe  therto  ,  then  *  t Archimedes  had  (for  finding  the  fraude  vfed  in  the 
Kinges  Crowne,  of  Gold) :  as  all  men  may  eafily  ludge  :  by  the  diuerfitie  of  the 
frute  following  of  the  one, and  the  other .  Where  I  fipakc  before,  of  a  hollow  Cu¬ 
bik  C offen.-the  like  vle,is  ofit:  and  without  waight.Thus.  Fill  it  with  water,  preci- 
fely  full,and  poure  that  water  into  your  Pyramis  or  Cone,  And  here  note  the  lines 
cutting  in  your  Pyramis  or  Cone .  Againe,fill  your  coffen,like  as  you  did  before. 
Put  that  Water,alfo,to  the  firft .  Marke  the  fecond  cutting  of your  lines .  N ow, 
as  you  proceded  before,  fo  muft  you  here  precede .  *  And  if  the  Cube,  which  you 
lb ou Id  Double,  be  ncuer  lb  great ;  you  haue,  thus,  the  proportion  (in  ftnall )  be- 
twene  your  two  litle  C  ubes :  And  then, the  fide, of  that  great  Cube(to  be  doubled) 
being  the  third  ,  will  haue  the  fourth,  found,  to  it  proportionall :  by  the  12.  of  the 
fixth  ofEuclide. 

N ote,that  all  this  while,”!  forget  not  my  firft  Propofition  Staticall,here  rehear- 
fed:  that,  the  Superficies  of  the  water, is  Spharicall .  Wherein,  vfe  your  diferetion: 
to  the  firft  line,addinga  finall  heare  breadth,more:and  to  the  fecond,halfe  a  heare 
breadth  more, to  his  length  .  For, you  will  eafily  perceaue,  that  the  difference  can 
be  no  greater,  in  any  Pyramis  or  Cone,  of  you  to  be  handled.  Which  you  fhall 
thus  try  e  .  F  or  finding  the fwelling  of  the  water  aboue  leuell .  Square  the  Semidiame¬ 
ter,  from  the  Centre  of  the  earth,to  your  firft  Waters  Superficies .  Square  then, 
halfe  the  Subtendent  ofthat  watry  Superficies  (  which  Subtendentmuft  haue  the 
equall  partes  of  his  meafure,  all  one,  with  thofe  of  the  Semidiameter  of  the  earth 
to  your  watry  Superficies)  :  Sub  trade  this  fquare,ffom  the  firft:  Of  the  refidue, 
take  the  Rote  Square.  That  Rote, Subtrade  from  your  firft  Semidiameter  of  the 
earth  to  your  watry  Superficies :  that,  which  remaineth,  is  the  heith  of  the  water, 
in  the  middle,  aboue  the  leuell .  Which, you  will  finde,  to  be  a  thing  infenfible. 
And  though  it  were  greatly  fenfible,*  yet,  by  helpe  of  my  fixtTheoreme  vpon  the 
laft  Propofition  of  Euclides  twelfth  booke,  noted :  you  may  reduce  all, to  a  true 
Leuell .  But,  farther  diligence,of you  is  to  be  vfed,againft  accidentall  caufes  of  the 
waters  fwelling:as  by  hauing(fomwhat)with  amoyft  Sponge,before,made  moyft 
your  hollow  Pyramis  or  Cone,  will  preuent  an  accidentall  caufe  of  Swelling,  &c. 
Experience  will  teach  you  abundantly :  with  great  eafe,  pleafure,and  comoditie. 

Thus,may  you  Double  the  Cube  Mechanically,  Treble  it,  and  fo  forth,  in  any 
proportion .  N ow  will  I  Abridge  your  paine,  coft,  and  Care  herein.  Without  all 
preparing  ofyour  Fundamentall  Cubes :  you  may  (alike)  worke  this  Conclufion. 
For, that, was  rather  akinde  of  Experimentall  dem6ftration,then  the  fhorteft  way: 

and 


lohn  Dee  his  Mathematical!  Er^face. 

and  all,  vpon  one  Mathematicall  Demonftration  depending  .  Take  water  (  as 
much  as  conueniently  will  ferue  your  turne :  as  I  warned  before  of  your  Funda¬ 
mental!  Cubes  bignes  )  Way  it  precifely .  Put  that  water, into  your  Pyramis  or 
Cone .  Of  the  lame  kinde  of  water,  then  take  againe,  the  lame  waight  you  had 
before :  put  that  fikewile  into  the  Pyramis  or  Cone .  For,  in  eche  time,  your  mar¬ 
king  of  the  lines,  how  the  Water  doth  cut  them,  Ibail  geue  you  the  proportion  be- 
twen  the  Radicall  fides,crf any  two  Cubes, wherof  the  one  is  Double  to  the  other: 
working  as  before  I  haue  taught  you:*lauing  that  for  you  Fundamental!  Cube  his 
Radicall  fide:  here,you  may  take  a  right  line,  at  pleafure. 

Yet  farther  preceding  with  ourdroppeof  Naturall  truth :  you  may  (now) 

geue  Cubes, one  to  the  other,  in  any  proportio  geue:  Rational!  orlr- 
rationall :  on  this  maner.Make  a  hollow  Parallelipipedon  of  Copper  or  Tinn'c: 
with  one  Bafe  wating,  or  open:as  in  our  Cubike  Coffen.  Fro  the  bottomc  of  that 
Paralielipipedon,raife  vp,many  perpendiculars, in  euery  ofhis  fower  fides.Now  if 
any  proportion  be  alfigned  you, in  right  lines. -Cut  one  ofyour  perpendiculars  (or 
aline  equall  to  it ,  or  lefie  then  it )  likewife :  by  the  lo.of  the  fixth  of Eu elide.  And 
thofe  two  partes ,  fet  in  two  fundry  lines  of thofe  perpendiculars  (  or  you  may  fet 
them  both, in  one  line)  making  their  bcginninges,to  be,  at  the  bale:  and  fo  their 
lengthes  to  extend  vpward  .  Now,  fet  your  hollow  Parallelipipedon,  vpright, 
perpendicularly,fteadie .  Poure  in  water,  handfomly,totheheith  ofyour  Ihorter 
line .  Poure  that  water,  into  the  hollow  Pyramis  or  Cone .  Marke  the  place  of 
the  riling.  Settle  your  hollow  Parallelipipedon  againe  .  Poure  water  into  it: 
vnto  the  heithofthe  fecond  line  ,  exa&ly  .  Poure  that  water  *  duely  into  the 
hollow  Pyramis  or  Cone :  Marke  now  againe,  where  the  water  cutteth  the  lame 
line  whichyoumarkedbefore  .  For,  there,  as  the  firft  marked  line,  is  to  thefe- 
cond  ;  So  fhall  the  two.  Radicall  fides  be,  one  to  the  other,  of  any  two  Cubes: 
which,  in  their  Soliditie,  lhall  haue  the  lame  proportion,  which,  was  at  the  firft  afi 
figned :  wereitRationali  or  Irrationall  .• 

Thus, in  fundry  waies  you  may  furnilhe  your  felfe  with  fuch  ftraungc  and  pro¬ 
fitable  matter:  which, long  hath  bene  wifhed  for.  And  though  it  be  Naturally  done 
and  Mechanically :  yet  hath  it  a  good  Demonftration  Mathematicall .  Which  is 
this  .•  Alwaies  ,you  haue  two  Like  Pyramids  :  or  two  Like  Cones,in  the  proporti¬ 
ons  alfigned :  and  like  Pyramids  or  Cones,are  in  proportion,one  to  the  other,  in 
theproportion  of  their  Homologall  fides  (or  lines)  tripled.  Wherefore,  if  to  the 
firft,  and  fecond  lines,foun  din  your  hollow  Pyramis  or  Cone,  you  ioyne  a  third 
and  a  fourth,  in  continuallpropbrtion  .-  that  fourth  line,  lhall  be  to  the  firft,  as  the 
greater  Pyramis  or  Cone,  is  to  the  lelfe :  by  the  33-of  the  eleuenth  ofEuclide  .  If 
Pyramis  to  Pyramis,  or  Cone  to  Cone,  be  double ,  then  lhall  *  Line  to  Line,  be 
alfo  double,  &c.  But,as  our  firft  line,  is  to  the  fecond, fo  is  the  Radicall  fide  of  our 
Fundamentall  Cube,to  the  Radicall  fide  of  the  Cube  to  be  made ,  or  to  be  dou¬ 
bled:  and  therefore, to  thofe  twaine  alfo,  a  third  and  a  fourth  line ,  in  continuall 
proportion,  ioyned :  will  geue  the  fourth  line  in  that  proportion  to  the  firft,as  our 
fourth  Pyramidall,  or  Conike  line,  was  to  his  firft :  but  that  was  double,  or  tre- 
ble,&c.as  the  Pyramids  or  Cones  were, one  to  an  other  (as  we  haue  proued)  thcr- 
fore,this  fourth,  ihalbe  alfo  double  or  treble  to  the  firft,as  the  Pyramids  or  Cones 
were  one  to  an  other  .*  But  our  made  Cube,is  deferibed  of  the  fecond  in  proporti¬ 
on, of  the  fower  proportionall  lines :  therfore  *  as  the  fourth  line, is  to  the  firft,  lb 
is  that  Cube, to  the  firft  Cube :  and  we  haue  proued  the  fourth  line,  to  be  to  the 
firft,  as  the  Pyramis  or  Cone,  is  to  the  Pyramis  or  Cone Wherefore  the  Cube  is 

c.iij.  to  the 


n 

yy 

yy 

yy 

yy 

}} 

yy 

T ogme  Cubes 
one  to  the  o'— 
t her  in  any 
proportion. 
Ratio  nail  or 
Irrationall. 

yy 

yy 

>t 

it 

j? 

}> 


?> 

?> 

» 

yy 

» 


'  Empty¬ 
ing  the 
firft. 


The  demonfiratient 
vf this  Dubbling  of 
the  Cube  And  of  the 
reft. 


r.D. 

*  Hereby  ,helpe  you, 

filfto  become  a  pr, 
cife  praSHftr.  ^Ant, 
fo  con ftder.horo, no¬ 
thing  at  all .  you  ar 
hindred(ftnftbly ) , 
the  Conuexitie  of 
the  water. 


rBy  the  3!.cf  the  e 
leuenth  bookc  of 
Euclide . 


l.T>. 

*Undy  out  diligence 
in  praftije.can  Jo 
f  in  -tonight  ofiva- 
terjpeiforme  it: 
Therefore, now. you 
are  able  to geue good 
reafon  of  your  whole 
doing. 


* Note  this 
Corollary. 


*T he great 
Commodities 
following  of 

thefe  new  In¬ 
ventions. 


Such  is  the 
Fruiteofthe 
Mathemati - 
tali  Sciences 
and  Artes ♦ 


lohn Dee  his  Mathematicall  Preface. 

to  the  Cube, as  Pyramis  is  to  Pyramis, or  Cone  is  to  Cone .  But  we  *  Suppofc  Py- 
ramis  to  Pyramis,or  Cone  to  Cone,  to  be  double  or  treble.&c.  Therfore  Cube, is 
to  Cube,double,or  treble, &c.Which  was  to  be  demonftrated.  And  of  the  Paralle- 
lipipedo,it  is  euidet ,  that  the  water  Solide  Parallelipipedons,are  one  to  the  other, 
as  their  heithes  are,feing  they  haue  one  bale .  Wherfore  the  Pyramids  or  Cones, 
made  of  thofe  water  Parallelipipedons,are  one  to  the  other, as  the  lines  arefone  to 
the  other) betwene  which,our  proportion  was  affigned  .  But  the  Cubes  made  of 
lines, after  the  proportio  of  the  Pyramidal  or  Conik  homologall  lines, are  one  to  the 
other,as  the  Pyramides  or  Cones  are ,  one  to  the  other  (  as  we  before  did  proue) 
therfore, the  Cubes  made,  flralbe  one  to  the  other,as  the  lines  alfigned,are  one  to 
the  othen'Which  was  to  be  demonftrated.Note.  *This,my  Demonftratiois  more 
generall,then  onely  in  Square  Pyramis  or  Cone:  Confider  well .  Thus ,  haue  I, 
both  Mathematically  and  Mechanically, ben  very  long  in  wordesiyet  fl  truft)no- 
thing  tedious  to  thcm,who,  to  thefe  thinges ,  are  well  affedied.  And  verily  I  am 
forced (auoiding  prolixitiejto  omit  fundry  fuch  things, ealie  to  be  pradifed: which 
to  the  Mathematicien,would  be  a  great  Threafure  :  and  to  the  Mechanicien,no 
fmall  gaine.*N ow  may  you, Betwene  two  lines  ginen ,finde  two  middle 
proportionals, in  Continuall  proportion  :  by  the  hollow  Paralleli- 
pipedon,  and  the  hollow  Pyramis,. or  .Cone..  Now,anyParallelipipedon 
rectangle  being  giuen:thre  right  lines  may  be  found,proportionall  in  any  propor¬ 
tion  aiTigned,ofwhich,ilial  be  produced  a  Parallelipipedon,  a?quall  to  the  Paralle- 
lipipedon  giuen.Hereof,I  noted  fomwhat,vpon  the  36,propolition,ofthe  n.boke 
of Euclide.  N ow,all  thofe  thinges, which  Vitruuius  in  his  Archite<fture,fpecified 
liable  to  be  done,  by  dubbling  of  the  C  ube  Or,  by  finding  of  two-middle  propor- 
tionall  lines,  betwene  two  lines  giuen,may  eafely  be  performed  .  Now,thatPro- 
bleme,  which  I  noted  vnto  you, in  the  end  ofmy  Addition,  vpon  the  34.ofthe  n. 
boke  of  Euclide ,  is  proued  pofsible.  N  ow,may .  any  regular  body,  be  T  ranflormed 
into  an  other, &c.  N ow,  any  regular  body : any  Sphere,  yea  any  Mixt  Solid :  and 
(that  more  is)Irregular  Solides,  may  be  made(in  any  proportio  affigned)like  vnto 
the  body, firft  giuen.  Thus, of  a  Manneken,  (as  the  Dutch  Painters  terme  it)in  the 
fame  Symmetric ,  may  a  Giant  be  made.-  and  that, with  any  gefture,by  the  Manner 
ken  vfed :  and  contrary  wife.N  o  w,  may  you ,  of  any  Mould,  or  Model!  of  a  Ship, 
make  one,of  the  lame  Mould  (in  any  affigned  proportion)  bigger  or  lelfer.  Now, 
may  you,of  any*Gunne,or  little  peece  or  ordinauce,make  an  other,with  the  fame 
Symmetric  ( in  all  pointes)  as  great, and  as  little,as  you  will.Marke  thatr  and  thinke 
on  it ,  Infinitely ,  may  you  apply  this,  fo  long  fought  for, and  now  fo 
eafily  concluded :  and  withall,fo  willingly  and  frankly  communi¬ 
cated  to  fuch,  as  faithfully  deale  with  vertuous  ftudies.Thus,canthe 
Mathematicall  minde, deale  Speculatiuely  in  his  own  Arte:  and  by  good  meanes. 
Mount  aboue  the  cloudes  and  fterres :  And  thirdly, he  can, by  order, Defcend,to 
frame  Naturall  thinges,  to  wonderful!  vfes  :and  when  he  lift ,  retire  home  into  his 
owne  Centre :  and  there,prepare  more  Meanes, to  Afcend  or  Defcend  by :  and, 
all,to  the  glory  of  God ,  and  our  honeft  delegation  in  earth. 

Although,  the  Printer ,  hath  lookedfor  this  Preface, a  day  or  two ,  yet  could  I 
not  bring  my  pen  from  the  paper ,  before  I  had  giuen  you  comfortable  warning, 
and  brief  inftru<ftion  s,offome  of  the  Commodities,by  Statike^  hable  to  be  reaped: 
In  the  reft,I  will  therfore,be  as  brief, as  it  is  pofsible/and  with  all,defcribing  them, 
fomwhat  accordingly.  And  that,you  fhall  percciue,by  this,  which  in  order  com- 

v  ineth 


lohn  Dee  his  Mathematical!  Preface. 

meth  next.  For,wheras,  it  is  fo  ample  and  wonderfu  11, that, an  whole  yeare  long, 
one  might  finde fruitfull  matter  therin,to  fpeake  of:and  alfo  in  pra&ile3is  a  Threa- 
fure  endeles  :yet  will  I  glanfe  ouer  it, with  wordes  very  few. 

This  do  I  call  Anthropographte.  which  is  an  Art  reftored ,  and  of 
my  preferment  to  your  Seruice.  I  pray  you,  thinke  of  it ,  as  of  one  of  the  chief 
pointes,of  Humane  knowledge.  Although  itbe,butnow,firft  Cofirmed,  with  this 
new  name  :  yet  the  matter,  hath  from  the  beginning,  ben  in  confederation  of  all 
perfed  P  hilofophers.  Anthropographie,is  the  defcription  of  the  Num¬ 
ber  ,Meafure,  Waight ,  figure,  Situation,  and  colour  of  euery  diuerfe 
thing, conteyned  in  the  perfect  body  of  MAN :  with  certain  know¬ 
ledge  of  the  Symmetric ,  figure ,  waight ,  Characterization,  and  due 
locall  motion, of  any  parcell  of  the  fayd  body,  afsigned;  and  of  N fi¬ 
bers, to  the  fayd  parcell  appertainyng.  This,is  the  onepartofthe  Defini¬ 
tion, mete  for  this  place:  Sufficient  to  notifie,  the  particularitie,  and  excellency  of 
the  Arte:and  why  itis,  here ,  afcribed  to  the  Mathematical.  Yf  the  defcription 
of  the  heauenly  part  of  the  world,had  a  peculier  Art,called  o Ajlronomie If  the  de¬ 
fcription  of  the  earthly  Globe,  hath  his  peculier  arte,call  ed  Geographic.  If  the  Mat¬ 
ching  ofboth,hath  his  peculier  Arte, called  Cofmographic :  Which  is  the  Defcriptid 
of  the  whole,and  vniuerfall  frame  of  the  world :  Why  fliould  not  the  defcription 
of  him,who  is  the  Leife  world:and,fro  the  beginning,called  CVticrocofmtts ( that  is.  M 
JJT  he  Lefte  World.)kn&  for  whole  lake,  and  feruice,all  bodily  creatures  els,  were  t^oe  LcIfe 
created  :  Who,alfo,participateth  with  Spirites,  and  Angels :  and  is  made  to  the  I-  W&rld‘ 
mage  and  fimilitude  of  GW:  haue  his  peculier  Artc'and  be  called  the  frteofArtesi 
rather,  then,  either  to  want  a  name,or  to  haue  to  bale  and  impropre  a  name  i  You 
muft  offundry  profeffions,borow  or  challenge  home ,  peculierpartes  hereofrand 
farder  procede:  as, God,  Nature ,  Reafon  and  Experience  lhall  informe  you.  The 
Anatomiftes  willreftore  to  you,fome  part:  The  Phyfiognomiftes,fome:The  Chy- 
romantiftes  fome.The  Metapofcopiftes,fome:  The  excellent,  Albert  Durer,*.  good 
part: the  Arte  ofPerfpediue,will  fomwhat,for  the  Eye,helpe  forward :  Pythagoras , 
Hipocrates, Plato,Galenus,Meletius,&  many  other  (in  certaine  thinges  )  will  be  Con- 
tributaries.  And  farder, the  Heauen,the  Earth, and  all  other  Creatures, will  eche 
fiew, and  offer  their  Harmonious  feruice  ,  to  fill  vp, that, which  wanteth  hereof: 
and  with  your  own  Experience,  concluding :  you  may  Methodically  regifter  the 
whole, for  the  pofteritie :  Whereby,  good  profc  will  be  had,  of  our  Harmonious, 
and  Microcofinicall  conftitution.  The  outward  Image,and  vew  hereof:  to  the  Art 
of  Zographie  and  Painting,  to  Sculpture ,  and  'Archite&ure (for  Church,Houfe,  Mwocofi 
Fort, or  Ship)  is  moft  necefaiy  and  profitable :  for  that,  itis  the  chiefe  bafe  and  mm. 
foundation  ofthem  .  Lookein  *  Vitruuius, whether  I  deale  fincerely  foryour  *  Lib  y* 
behoufe,  or  no .  Lookein  Alhertus  D ur er us, De  Symmetriahumani  Corporis.  Looke  ^ap • 1  • 
in  the  2  7. and  28.  Chapters, of  the  lecond  booke,  De  occulta  Philofophia  .  Confi- 
der  th eArke  of  Hoe .  And  by  that,  wade  farther .  Remember  the  Delphicall  Oracle 
2i0  S  C  E  T  El  P  S  v  M  (  Kpnoive  thy  felfe  )  fo  long  agoe  pronounced :  of fo 
many  a  Philofopher  repeated :  and  of  the  Wifett  attempted :  And  then,  you  will 
perceaue,  how  long  agoe,  you  haue  bene  called  to  the  Schole,where  this  Arte 
might  be  learned.  Well.  Iam  nothing  affrayde,ofthedildayneoflbme  Rich,  as 
thinke  Sciences  and  Artes,  to  be  but  Seuen.  Perhaps, thole  Such, may,  with  igno¬ 
rance,  and  lhame  enough,  come  fliort  ofthem  Seuen  alio  :  and  yet  neuerthelefe 

c.iiij.  they 


lohn  Dee  his  Mathematical!  Preface, 

they  can  not prcfcribe  a  certaine  number  of Artesrand  in  eche, certaine  vnpaflable 
boundes,ro  God,Naturc,and  mans  Induftrie.New  Artes,dayly  rife  vp:  and  there 
O*  was  no  fuch  order  taken, that,  All  Artes,lbould  in  one  age, or  in  one  land, or  of  one 

man,be  made  knowen  tothe  world.Let  vs  embrace  the  giftes  of  God ,  and  wayes 
to  wifedome ,  in  this  time  of  grace ,  from  aboue ,  continually  bellowed  on  them., 
who  thankef  ully  will  receiue  them :  Et  bonis  Omnia  Ccoperabuntur in  bonum. 

Trochilike,  is  that  Art  Mathematical!, which  demonftrateth. 
theproperties  of  all  Circular  motions ,  Simple  and  Compounde. 
And  bycaufe  the  frute  hereofvulgarly  receiued,is  in  Wheles ,  it  hath  the  name  of 
Trochilike :  as  a  man  would  lay ,Whele  Art. By  this  art, a  Whele  may  be  geuen  which 
lhall  moue  ones  about,  in  any  tyme  aligned  .  Two  Wheles  may  be  giuen, 
whofe  turnynges  about  in  one  and  the  fame  tyme,  (  or  equall  tymes)  ,  lhall  haue, 
one  to  the  other,  any  proportionappointed .  By  Wheles,  may  a.  ftraight  line  be 
defcribed :  Likewife,a  Spirail  line  in  plaine,Conicali  Se&ion  Iines,and  other  Irre¬ 
gular  lines,  at  pleafure,  may  be  drawen .  Thefe,  and  fuch  like,  are  principal!  Con- 
clulions  of  this  Arte  :  and  helpe  forward  many  plealant  and  profitable  Mechani- 
Saw  Milks,  call  workcs  :  As  Milles,to  Saw  great  and  very  long  Deale  bordes ,  no  man  being 
by .  Suchhaue  I  feenein  Germany :  and  in  the  Citieof  Prage ;  in  thekingdome 
of  Bohemia :  Coyning  Milles,Hand  Millesfor  Corne  grinding:  Andallmaner  of 
Milles,and  Whele  worke:  By  Winde,  Smoke,  Water,  Waight,  Spring,  Man  or 
Beall, moued .  Take  in  your  hcmd,Agricola  here  CAEctalUca  :  and  then  lhall  you 
(in  all  Mines)  perceaue,how  great  nede  is,  ofWhele  worke.By  Wheles,llraunge 
workes  and  in  credible, are  done  -  as  will, in  other  Artes  hereafter,  appeare.  A  won. 
derfull  example  of  farther  polfibilitie,  and  prefent  commoditie ,  was  fene  in  my 
time,  in  a  certaine  Inllrument:  which  by  the  Inuenter  and  Artificer(before)  was 
folde  for  xx.  Talentes  of  Golde:and  then  had  (by  milFortune)receaued  fome  iniu- 
rie  and  hurt :  And  one  lanellm  of  Cremona  did  mend  the  lame,  and  prefented  it  vn- 
to  the  Emperour  Charles  the  fifth .  Hieronymus  Cardanus ,  can  be  my  witnelle,  that 
therein,  was  one  Whele,  which  moued,  and  that,in  fuch  rate,thar,in  yoco.yeares 
onely,  his  owne  periode  fliould  be  finifiied .  A  thing  almoffc  incredible :  But  how 
farre,l  keepe  me  within  my  boundcs:  very  many  men(yetaliue)  can  tell. 

HellCofophlC,  is  nere  Siller  to  Trochilike:  and  is.  An  Arte  Mathema¬ 
tically  which  demonftrateth  the  defigning  of  all  Spirail  lines  in 
Plaine ,  on  Cylinder ,  Cone ,  Sphere ,  Conoid  ,  and  Spheroid,  and 
their  properties  appertayning  .  The  vfe hereof,  in  Architecture ,  and  di- 
uerfe  Inflrumentes  and  Engines, is  moll  necellary.  For,in  many  thinges,the  Skrue 
worketh  the  feate,  which,  els, could  not  be  performed  .  By  helpe  hereof ,  it  is 
*Atheneus  *  recorded,  that, where  all  the  power  of  the  Citie  of  Syracufa,was  not  hableto 
Ltb.  5  .cap, 2.  moue  a  certaine  Shipfbeing  on  ground)mightie  Archimedes , letting  to ,  his  Skruifh 
Engine ,  caufed  Hiero  the  king ,  by  him  felf ,  at  eafe,to  remoue  her,  as  he  would. 
Trodu.  Wherat,the  King  wondring :  A?ro mans acm-os,  ApyjAi  Aycm-msAo/s. 
Pag.  r  8 .  From  this  day,  forward  (laid  the  King  )  Credit  ought  to  be  giuen  to  Archimedes ,  what 
focuer  he  fayth. 

Pneumatithmie  demonflrateth  by  clofe  hollow  Geometri¬ 
cal!  Figures,(regular  and  irregular  )  theftraunge  properties  ( in  mo¬ 
tion  or  ftay)of  the  Water, Ayre, Smoke ,  and  Fire,in  theyr  cotinuitie, 

and 


Iohn  Dee  his  Mathematical!  Preface. 

and  as  they  are  ioyned  to  the  Elementes  next  them.  This  Arte ,  to  the 
Naturall  Philofopher,is  very  profitable:  to  proue,that  Vacuum ,  or  Emetines  is  not 
in  the  world.  And  that, all  N attire,  abhorreth  it  fo  much :  that ,  contrary  to  ordi¬ 
nary  law,the  Elementes  will  moue  or  hand .  As, Water  to  afcend: rather  then  be- 
twene  him  and  Ayre,Space  or  place  lhould  be  left,mOre  then  (naturally)  that  qua- 
titie  of  Ayre  requireth,or  can  fill.  Againe,  Water  to  hang, and  not  defcend: rather 
then  by  defcending,to  leaue  Eiriptines  at  his  backe  .  The  like,  is  of  Fire  and  Ayre: 
they  willdefcend.-when,cither,  their  Cotinuitiefhould  be  dif!blued:or  their  next 
Element  forced  from  them.  And  as  they  willnotbeextended,todifcontinuitie: 
So, will  they  not, nor  yet  of  mans  force,canbe  preft  or  pent,in  fpace ,  notfufhcient 
and.aunfwcrable  to  their  bodily  fubftance.Great  force  and  violence  will  they  vfe, 
to  enioy  their  naturall  right  and  Iibertie.  Hereupon,  two  or  three  men  together, 
bykeping  Ayrevnder  a  great  Cauldron,  and  forcyng  the  fame  downe,  orderly, 
may  without  harme  defcend  to  the  Sea  bottome :  and  continue  there  a  tyme  &c. 
Where,N ote,how  the  thicker  Element(as  the  Water)giueth  place  to  the  thynner 
(as, is  the  ayre:)and  receiueth  violence  of  the  thinner, in  maner.  Sec.  Pumps  and 
all  maner  of  Bello  wes,  haue  their  ground  of  this  Art:  and  many  other  ftralinge  dc- 
uifes.  Asflydraulica&rganes  goyng  by  water.  Sec.  Of  this  Feat,  (called  common¬ 
ly  Pneumatica> )  goodly  workes  are  extant ,  both  in  Greke,and  Latin .  With  old 
and  learned  S  chole  m en,it  is  called  S dentin  de pleno  &  vacuo. 

Menadrie,  isan  Arte  Mathematicall, which  demonfirateth, 
how ,  aboue  Natures  vertue  and  power  fimple :  Vertue  and  force 
may  be  multiplied  :  and  fo,  to  dired,to  lift,  to  pull  to ,  and  to  put  or 
caft  fro  ,  any  multiplied  or  fimple ,  determined  Vertue ,  Waight  or 
Force:  naturally, not, fo ,  diredible  or  moueable.  Very  much  is  this  Art 
furdred  by  other  Artes  •  as,  in  fome  poihtes,  by  Perjfeciiue:  in  fome,  by  St  at  ike :  in 
fome,by  Trochili  ke.znd  in  othcr,by Helicofophie.-znd  Pneumatithmie.  By  this  Art, 
all  Cranes, Gybbettes,&  Ingines  to  lift  vp ,  or  to  force  any  thing, any  maner  way, 
are  ordred :  and  the  certaine  caufe  of  their  force, is  knoWne :  As,  the  force  which 
one  man  hath  with  the  Duche  waghen  Racke:therwith,to  fet  vp  agayne,a  mighty 
waghenladen,befng  ouerthrowne.  The  force  of  the  Croflebow  Racke,  is  certain¬ 
ly,  here,demonftrated.The  reafon,why  one  m5,  doth  with  a  Ieauer,Iift  that8which 
Sixe  men,  with  their  handes  onely,  could  not,  fo  eafily  do.  By  this  Arte,in  ottr 
common  Cranes  in  London ,  wherepowreisto  Cranevp,thewaight  df  2000. 
pound: by  two  Wheles  more  (by  good  order  added  )  Arte  concluded!,  that  there 
may  be  Craned  vp  20oooo.pound  waight  &c.So  well  knew  Archimedes  this  Arte.* 
that  he  alone, with  his  deuifes  and  engynes,(twife  or  thrife)fpoyIed  and  difeomfi- 
ted  the  whole  Army  and  Hofte  of  the  Romaines,  befieging  Syracufa,  Marcus  Mar- 
cellus  the  Confed,  being  their  Generali  Capitaine.  Such  huge  Stones,  fo  many, with 
fuch  force ,  and  fo  farre ,  did  he  with  his  engynes  hay  le  among  them,  out  of  the 
Citie.  And  by  Sea  likewife :  though  their  Ships  might  come  to  the  walls  ot'Syra- 
cufa ,  yet  hee  vtterly  confounded  the  Romaine  Nauye.  >  What  with  his  mighty 
Stones  hurlyng:  what  with  Pikes  of*  /  8  fote  long, made  like  lhaftes:  which  he  for¬ 
ced  almoft  a  quarter  of  a  myle.What,  with  his  catchy  ng  hold  of  their  Shyps  ,  and 
hoyfing  them  vp  aboue  the  water  ,  and  fuddenly  letting  them  fail  into  the  Sea  a- 
gaine:what  with  his*  Burning  Glades; by  which  he  fired  their  other  Shippcs  afar- 
of:  what, with  his  other  pollicies, deuifes,  and  engines,  he  fo  manfully  acquit  him 
felfe :  that  all  the  Fotce,courage,and  pollicie  of  the  Romaines  (for  a  great  feafon) 

d.j.  could 


To  goto  the 
bottom  of  the 
Sea  Without 
damoer. 

o 


Plutarchus  in  Mir* 
eo  Marcello . 
Synejiusia  Epifio * 
lis. 

Polybius* 

Plinius. 

QuintilUnus. 

T.  Liuius. 
*iAthen&uf, 


*  Qalenus. 

o 

xAnthemtus. 


‘Burning 

GLtJfes. 


Games* 


lohn  Dee  his  Matheifiatkall  Preface . 

could  nothing  prcuailc/or  the  winning  ofSyracula.  Wherupon ,  the  Romanes 
named  Archimedes, Briareus^nd  Cmtimmus.  Zonaras  maketh  mention  of  one  Pro-* 
clftSj who  fo  well  had  perceiued  Archimedes  Arte  of  c Memdrie ,  and  had  fb  well  in* 
uented  of  his  owne  ,  that  with  his  Burning  Glades,  being  placed  vpon  the  walks 
of  Byfance ,  he  multiplied  lb  the  heate  of  the  Sunne,and  direded  the  bcames  of 
the  lame  again  ft  his  enemies  Nauie  with  fuch  force ,  and  fo  fodeinly  ( like  lighte- 
ning)that  he  burned  and  deftroyed  both  man  and  ihip .  And  Dion  fpecifieth  of 
Prijcits, a  Geometrickn  in  Bylance,  who  inuented  and  vied  fondry  Engins,  of  Force 
multiplied :  Which  was  caule,  that  the  Empcroxr  Seuerus  pardoned  him,  his  lifc,af* 
ter  he  had  wonne  BylancecBycaufe  he  honored  the  Arte ,  wytt,  and  rare  induftrie 
of P  rife  ft s.  But  nothing  inferior  to  the  inuention  of  thefe  engines  of  Force,  was  the 
inuention  ofGunnes.  Which,  from  an  Englifh  man, had  the  occalion  and  order 
of  firft  inuenting:  though  in  an  other  Iand,and  by  other  men,it  was  firft  cxecuted.- 
And  they  thatlhould  fee  the  record,  where  the  occalion  and  order  general!,  of 
3)  Gunning, is  firft  difcourfed  of,wouid  thinke:  that,linall  thinges,llight5and  comon : 
3,  commingto  wife  mens  confideration,and  induftrious  mens  handling ,  may  grow 
,3  to  be  of  force  incredible. 

Hypogciodie,  is  an  Arte  Mathematical!,  demonftratyng,how, 
vnder  the  Sphxricall  Superficies  of  the  earth,  at  any  depth,  to  any 
perpendicular  line  afsigned(whofe  diftance  from  the  perpendicular 
of  the  entrance:  and  the  Azimuth  ,likewife, in  refped:  of  the  faid  en¬ 
trance,  is  knowen)  certaine  way  may  be  prarfcribed  and  gone :  And 
how,  any  way  aboue  the  Superficies  of  the  earth  deligned ,  may  vn~ 
d^r  earth, at  any  depth  limited ,  be  kept  :  goyng  alwayes ,  perpendi¬ 
cularly  ,vnder  the  way  ,  on  earth  defigned  :  And,  contrarywife,Any 
way,(ftraight  or  croked ,  )vnder  the  earth,  beynggiuen  :  vppon  the 
vtface, or  Superficies  of  the  earth, to  Lyne  out  the  fame :  So, as,  from 
the  Centre  of  the  earth  ,  perpendiculars  drawen  to  the  Spherical! 
Supefficies  of  the  earth,  fhail  precifely  fall  in  the  Correfpondent 
poirites  of  thofe  two  wayes  .  This ,  with  all  other  Cafes  and  cir- 
cumftances  herein ,  and  appertenances ,  this  Arte  demonftrateth . 
This  Arte,  is  very  ample  in  vanetie  of  Conclusions :  and  very  profitable  fundry 
wayes  to  the  Common  Wealth .  The  occafion  of my  Inuenting  this  Arte, was  at 
the  requeft  of  two  Gentlemen, who  had  a  certaine  worke(  of  gaine)  vnder  ground: 
and  their groundes  did  ioyne  ouer  the  worke :  and  by  reafon  of  the  crokednes, 
diuers  depthes,  and  heithes  of  the  way  vnder  ground ,  they  were  in  doubt,  and  at 
controuerfie,  vnder  whofe  ground,  as  then,  the  worke  was  *  The  name  onely  (be¬ 
fore  this  )  was  of  me  publifhed,  Deltinere  Snbtcrraneo  The  reft, be  at  Gods  will. 
For  Pioners,  Miners,  Diggers  for  Mettalls,  Stone,  Cole,  and  for  fecretepaffages 
vnder  ground, betwene  place  and  place  (as  this  land  hath  diuerie)  and  for  other 
purpofes^any  man  may  eafily  perceaue,  both  the  great  fruite  of  this  Arte,  and  alfo 
in  this  Arte,  the  great  aide  of  Geometric. 

Hydragogie,  demonftrateth  the  poffible  leading  of  Water, by 
Natures  lawe  ,  and  by  artificial  helpe  ,  from  any  head  (being a 

Spring,  {landing,  or  running  Water  )  to  any  other  place  affigned. 

Long 


Iohn  Dee  his  Mathematical!  Preface, 

Long, hath  this  Arte  bene  in  vfe :  and  much  thereof  written :  and  very  marueilous 
workes  therein, performed :  as  may  yetappeare,in  Italy :by  the  Kuynes  remaining 
of  the  Aquedu&es .  In  other  places, of  Riuers  leading  through  the  Maine  land* 

Nauigable  many  a  Mile.  And  in  other  places,of  the  marueilous  forcinges  of  Wa¬ 
ter  to  Afcend .  which  all, declare  the  great  fkiIl,to  be  required  ofhim,who  fhould 
in  this  Arte  be  perfe&c,  for  all  occafions  of  waters  poftlble  leading .  T  o  Ipeakc 
of  the  allowance  of  the  Fall, for  euery  hundred  foote:  or  of  the  Ventills  (if  the  wa¬ 
ters  labour  be  farre,and  great)  I  neede  not  .•  Seing,  at  hand  (about  vs)many  expert 
men  can  fufficiently  teftific,  in  effe&c,  the  order  :  though  the  Demonftration  of 
the  N  eceifitie  thereof*  they  know  not  :  N  or  yet ,  if  they  lhould  be  led*  vp  and 
downe,  and  about  Mountaines,  from  the  head  of  the  Spring:  and  then, a  place  be¬ 
ing  afligned :  and  of them,  to  be  demaunded,  how  low  or  high,  that  laft  place  is,  in 
refpe&e  of  the  head,  from  which  (fo  crokediy,  and  vp  and  downe  )  they  be  comes 
Perhaps,they  would  not,  or  could  not,  very  redily,or  nerely  afloyle  that  quefrion. 

Geometric  therefore,  is  neceflary  to  Hydragogie .  Of  the  fimdry  wayes  to  force  wa¬ 
ter  to  afcend ,  eyther  by  T ympane,  Kettell mills,  Skrue ,  Cteftbike ,  or  filch  like :  in  Vi~ 
truuim,  ^Agncola,  (and  other,) fully, the  maner  may  appeare .  And  fo, thereby ,alfb 
be  m oft  euident,  how  the  Artes,  of  P neumatithmie, Heiicofophie,  Statike ,  Trochihke , 
and  CMenadrie ,  come  to  the  furniture  of  this,in  Speculation,  and  to  the  Corn  mo¬ 
di  tie  of  the  Common  Wealth,in  praftife. 

Horometrie,  is  an  Arte  Mathematical!,  which  demofbrateth, 

how, at  all  times  appointed,  theprecife  vfuall  denominatio  of  time, 
may  be  knowen, for  any  place  afligned .  Thefe  wordes,are  finoth  and 
plaine  ealie  Englilhe,  but  the  reach  of  their  meaning, is  farther,  then  you  woulde 
lightly  imagine .  Some  part  of  this  Arte,  was  called  in  olde  time,  Gnomomce:  and 
of  lat  eflorologiographia :  and  in  Englilhe, may  be  term  ^Dulling  .  Auncient  is 
the  vfe ,  and  more  auncient,is  the  Inuention .  The  vfe,doth  well  appeare  to  haue 
bene  (at  the  Ieaft)  aboue  two  thoufand  and  three  hundred  yeare  agoe in  *  King  ^Reg.zo* 
K^dchaz,  Diall,  then,by  the  Sunne,lhewing  the  diftin&ion  of  time .  By  Sunne, 
Mone,and  Sterres,this  Dialling  may  be  performed, and  the  precife  Time  of  day  or 
night  kno  wen .  But  the  demonftratiue  delineation  of  thefe  Dialls,of  all  fortes* 
requireth  good  fkill, both  of  AHronomie, and  Elementall, SphccricalfPhx- 

nomenaII,and  Conikall .  Then, to  vfe  the  groundes  of  the  Arte,  for  any  regular 
Superficies,  in  any  place  offred :  and  ( in  any  poflible  aptpofition  therof)  theron, 
to  defcribe  (  all  maner  of  wayes )  how,  vfualLho  wers,  may  be  (  by  the  $  times  flia- 
dow  )  truely  determined :  will  be  found  no  Height  Painters  worke .  So  to  Paint* 
and  prefcribe  the  Sunnes  Motion, to  the  breadth  ofaheare.  In  this  Feate(in  my 
youth )  I  Inuented  a  way, How  in  any  Horizontall,Murall,or  ./Equine- 
dtiall  Diall, &c.  At  all  howers (the  Sunne  fhining)  the  Signe  and  De- 

gree  afcendent,may  be  knowen .  which  is  a  thing  vetf  neceffary  for 
the  Riling  of  thole  fixed  Sterres :  whofe  Operation  in  the  Ayre,  is  of  great  might, 
euidently .  I  Ipeake  no  further, of  the  vfe  hereof.  But  forafmuch  as, Mans  affaires 
require  knowledge  of  Times  &  Momentes,when,neither  Sunne,Mone,or  Sterre, 
can  be  fene:  Therefore, by  Induftrie  Mechanical!,  was  inuented,firft,how,by  Wa- 
ter,running  orderly, the  Time  and  howers  mightbe  knowen:  whereof,  theVamous 
Cteftbms ,  was  Inuentor :  a  man,  of Vitruuius,  to  the  Skie  (iuftly)  extolled .  Then, 
after  that,  by  Sand  running,were  howers'  meafured  :  Then,bv  TrocMke  with 
waight :  And  of  late  time,  by  Trochilike  with  Spring :  without  waight.  All  thefe, 

d.ij.  by- 


lohn  Dee  his  Mathematical!  Preface  . 

by  Sunne  or  Sterres  diredion  ( in  certaine  time )  require  ouerfight  and  reformati¬ 
on,  according  to  the  heauenly  dsquinodiall  Motion:  befidesthe  ineequalitie of 
their  owne  Operation .  There  remayneth  (without  parabolicall  meaning  herein) 
Afcrfctuall  among  the  Philofophers,a  more  excellent,  more  commodious,and  more  maruei- 
lous  way,  then  all  thefe  .*  ofhauing  the  motion  of  the  Primouant  (or  firft  a?quino- 
diall  motion, )by  Nature  and  Arte,Imitated:  which  you  fhall  (  by  furder  fearch  in 
waightier  ftudyes  )  hereafter, vnderftand  more  of.  And  fo,  it  is  tyme  to  firiifh  this 
Annotation,of  Tymes  diftindion,vfed  in  our  common,and priuate  affaires:  The 
commoditie  wherof,no  man  would  want, -that  can  tell,  how  to  beftow  his  tyme. 

Zograpllie^is  an  Arte  Mathematicall, which  teacheth  and  de- 
monftrateth ,  how ,  the  Interfedlion  of  all  vifuall  Pyramides ,  made 
by  any  playne  al signed,  ( the  Centre,  diftance,and  lightes,beyng  de¬ 
termined)  may  be,  bylynes,and  due  propre  colours,  reprefen  ted. 
A  notable  Arte, is  this*and  would  require  a  whole  Volume, to  declare  the  proper¬ 
ty  thereof:  and  the  Commodities  enfuyng .  Great  fkili  of  Geometric,  LsArithme- 
uke^PerJpeclme,  and  Antkropographie,  with  many  other  particular  Artes,hath  the  Zo¬ 
grapher,  nede  of  for  his  perfedion.For,  the  moft  excellent  Painter, (who  is  but  the 
propre  Mechanicien,  &  Imitator  fenfible,  of  the  Zographer)  hath  atteined  to  fuch 
perfedion,that  Senfe  of  Man  and  beaft,haue  iudged  thinges  painted,  to  be  things 
naturalfand  not  artificial!  :aliue, and  not  dead.This  Mechanicall  Zographer  ('com¬ 
monly  called  the  Painter)is  meruailous  in  his  fkill:and  feemeth  to  haue  a  certaine 
diuine  power:  As,offrendes  abfent,to  make  a  frendly ,  prefent  comfort :  yea,  and 
of  frames  dead, to  giue  a  continual! ,  filent  prefence :  not  onely  with  vs ,  but  with 
our pofteritie, fomany  Ages.  Andfoprocedyng,  Confider,How,in  Winter, he 
can  fhew  you, the  liuely  vew  of  Sommers  Toy, and  riches:and  in  Sommer,exhibite 
the  countenance  of  Winters  dolefu.U  State, and  nakednes.Cities,Townes, Fortes, 
Woodes,  Armyes, yea  whole  Kingdomes  (be  they  neuerfofarre,or  greate)  can 
he,with  eafbjbring  with  him,  home  (to  any  mans  Judgement )  as  Paternes  liuely, 
of  the  thinges  rehearfed.  In  one  little  houfe,  can  he,enclofe(with  great  pleafure 
of  the  beholders,)  the  portrayture  liuely,of  all  vifible  Creatures,either  on  earth,or 
in  the  earth, liuitig:  or  in  the  waters  lying,Creping,fiyding,or  fwimmingior  ofany 
foule,or fly, in  the  ayre  flying.  Nay, in  refped of  theStarres,the Skie,the  Cloudes: 
yea,in  the  fliew  of the  very  light  it  felfe  (that  Diuine  Creature  )  can  he  match  our 
eyes  Iudgement,moft  nerely.  Whata  thing  is  this'thinges  notyet  being,he  can 
reprefent  fo ,  as, at  their  being, the  Picture  fhall  feame  (in  maner)to  haue  Created 
them.  To  what  Artificer, is  not  Pidure,a  great  pleafure  and  Commodities  Which 
of  them  all, will  refufe  the  Diredion  and  ayde  ofPidure'The  Archited,the  Gold- 
fmith'and  the  Arras  Weauer :  ofPidure,make  great  account.  O  ur  liuely  Herbals, 
our  portraitures  ofbirdes,  beaftes,ancl  fifties  :  and  our  curious  Anatomies,which 
way,are  they  moll:  perfedly  made,or  with  moft  pleafiire,ofvs  beholden?  Is  it  not, 
by  Pidure  onely?  And  if  Pidure ,  by  the  Induftry  of  the  Painter,  be  thus  commo¬ 
dious  and  meruailous:  what  fhall  be  thought  of Zographie, the  Scholemafter  ofPi- 
dure,and  chief  gouernor?  Though  I  mention  not  Sculpture ,in  my  T able  ofArtes 
Mathematicall :  yet  may  all  men  perceiue,How,that  Picture  and  Sculpture ,  are  Si¬ 
fters  germaine:and  both, right  profitable ,  in  a  Commo  wealth.and  of  Sculpture, zf- 
well  as  of  Pitiure,excellent  Artificers  haue  written  great  bokes  in  commendation. 
Witnefle  I  take, of  Georgia  Vafarijittore  Aretino:of  Pomponius  Gauricus :  and  other. 
To  thefe  two  Artes,  (with  other,  )is  a  certaine  od  Arte ,  called  Althalmafat,  much 
beholdyng:  more,  then  the  common  Sculptor ,Entayler,Keruer,  Cutter, Grauer,  Foun - 


Iohn  Dee  his  Mathematical!  Preface. 

der,  or  Paynter(&c)knovf  their  Arte,to  be  commodious. 

Ar chltecture^to  many  may  feme  not  worthy,  or  not  mete,  to  be  reckned  -An  obiettion, 
among  the  Artes  UVtathematicall.To  whom, I  thinke  good, to  giue  fome  account  of 
my  fb  doyng.N'ot worthy,  (will  they  fay,)bycaufe  it  is  but  for  building, of  a  houfe, 

Pallace, Church, Forte, or  fuch  like,groffe  Workes.And  you,alfo,  defined  the  Artes 

OWathanaticalljLQ  be  fuch, as  dealed  with  no  Materiallor  corruptible  thing: and  al- 

fo  did  demonftratiuely  procedein  their  faculty  ,by  Number  or  Magnitude .  Firft, 

you  fee, that  I  covmt fetdyArchiteclurey  among  thole  ^Artes  Mathematically  which  The  Aufwer, 

are  Deriued  from  the  Principals :  and  you  know ,  that  fuch,may  deale  with  Na- 

turall  thinges,and  fenfible  matter ,  Of  which ,  fome  draw  nerer,to  the  Simple  and 

abfolute  Mathematicall  Speculation, then  other  do  .  And  though, the  Architect  ” 

procureth,  enformeth,  &  diredeth,the  Mechanicienf. o  handworke,  &  the  building  ” 

aduall,ofhoufe,Caftell,or  Pallace,  and  is  ehiefludgeofthefame :  yet ,  with  him 

felfe  (as  chief  CWafier  and  Architect  >  )  remaineth  the  Demonftratiue  reafon  and 

caufe,  of  the  Mechaniciens  worke.-  in  Lyne,plaine,  and  Solid :  by  Geometrically  A-  ” 

rithmeticall,Opticall,Muficall,AttronomicallyCofmographicall  ( &  to  be  brief)  by  all  the  ” 

former  Deriued  Artes  Mathematically  and  other  Naturall  Artes,hable  tot>e  confir-  ** 

med  and  ftablifhed.If  this  be  fo:  then,  may  you  thinke, that  Architecture, hath  good 

and  due  allowance ,  in  this  honeft  Company  of  Artes.  CMathematicall  Deriuatiue. 

I  will,herein,craue  Iudgement  of  two  moft  perfed  Architefies :  the  one ,  being  Vi- 
truuius,  the  Romaine :  who  did  write  ten  bookes  thereof, to  the  Emperour  ^Augu¬ 
stus  ( in  whofe  daies  our  Heauenly  Archemafter,  was  borne  )  :  and  the  other,  Leo 
Baptifla  Alhertus ,  a  Florentine :  who  alfo  published  ten  bookes  therof .  oArchi- 
tetlura  ( fayth  Vitruuius )  est  Scientia  pluribus  dijciplinis  dr  ruarijs  eruditionibus  or n  at  as 
cuius  Iudicio  probantur  omnia ,  qua  ab  cater  is  Artifcibus  perficiuntur  opera  .  That  is. 
Architecture,  is  a  Science  garnifhed  with  many  doCtrines  &  diuerfe 
inftruCtions :  by  whofe  Iudgement,  all  workes,by  other  workmen 
finifhed,  are  Illdged  .  It  followeth.&z  nafcitur ex  Fabrica ,  dr  Ratiocinatione.&e. 

Ratiocinatio  auttm  eft,  qua,res fabrtcatas,S  olertia  ac  ratione  proportions >demonBr are  atifi 
exphcare  potest .  jdrchiteffiure  groweth  ofFramingfind  %eafoning.i?c.  %ea* 
foningyis  ibdifwnch  of  t  hinges framed fiicith forecaft ,and proportion:  can  make 
demonstration^  and  manif eft  declaration  .  Againe.  Cum ,  in  omnibus  enim  re¬ 
bus,  turn  maxim l  etiam  in  Architectura,  hac  duo  infunt :  quod  fignifcatur,  dr  quod  figni- 
ficat .  Significant propofit a  res,  de  qua  dicitur  rhanc  autem  Si^mficat  Demonfiratioyrati- 
onibus doctrinarum explicata  .  Forafmuch  as  ,  in  all  t  hinges*  therefore  chiefly 
in  jdrc  hit  effur ey  t  hefe  two  thinges  are  :the  thing  fignifted:  and  that  Oflicb  fig • 
nifieth ,  The  thing  propounded ,  thereof  Wjpedke,  is  the  thing  Signified. 

But  ■Demonftr ation  ,expreffed  Toith  thereafonsof diuerfe  doSirines  ftoth  Jigni* 

fie  the  fame  thing .  After  that  .Ft  liter aius fit ,  peritus  Graphidos,eruditus  Geometria , 
dr  Optices  non  ignarus  :  infiructus  Artthmetica-.hifiorias  complures  nouerit ,  Philofophos 
dmgenter.audiuerit:%Muficamftiuerit  \  Medicina  non  fit  ignarus,  refionfa  lurijperitoru 
nouerit:  Afirologiam,  Calife  ratipn.es  cognitas  habeat  .An  Architect  (fayth  he)  ought 
to  ynderftand  Languages  yto  be  fkilfull  ofF  ainting ,  mellinftruffied  in  Geome * 
trie,  not  ignorant  ofFerJpeffiiue , furnijhed'toith  Arithmetike  faue  knowledge 
of  many  hiftories,  and  diligently  haue  hard  Fhilofophers y  haue  f  kill  of  Mu> 
fike,  not  ignorant  ofFlyfike ,  know  the  aunfweres  of  Lawyers ,and  haue  Aftro - 

d.itj.  nomiey 


lohn  Dee  his  Mathematical!  Preface. 

nomie,  and  the  courfes  C&lefliall ,  in  good  knowledge .  He  geueth  reafon  5  or¬ 
derly,  wherefore  all  thefe  Artes,Do&rines,and  Inftru&ions,  are  requisite  in  an  ex¬ 
cellent  Architect .  And  (for  breuitie)  omitting  the  Latin  text, thus  he  hath. 
Secondly,  it  is  behofefull for  an  Architedto  haue  the  knowledge  of  Taint  in?: 
that  he  may  the  more  eafilie fajhion  out fin patternes painted ,the forme  of  what 
Worke  he  liketh.  And  Geometrie,geuethto  Architecture  manyhelpes :  and first 
teacheth  the  Vfe  of the  flute,  and  the  Cumpafie:  wherby  (chiefly  and  eafilie )  the 
defcriptions  of  Buildinges ,  are  defpatched  in  Groundplats:  and  the  directions  of 
S  quires  ,Leuells  , and  Lines.  Likewifefiy  TerJpeCtiue/he  Light es  of the  be  a* 
nen,are  1 Kell  led  Jin  the  buildinges :  from  certaine  quarters  of  the  World .  (By 
Arithmetike  ,the  charges  of  Buildinges  are fiummed  together:  the  meafures  are 
exprefied,  and  the  hard  questions  of  Symmetries,  are  by  Geome tricall  Meanes 
and  Methods  difcourjed  on.  tyre.  Befides  this ,of the  Nature  of  tbinges( which 
in  Greke  is  called  ?u<7ioXo/i&  )  Thilojophie  doth  make  declaration  .  Which,  it  is 
neceffary ,for  an  Architect, with  diligence  to  haue  learned :  becaufe  it  hath  ma • 
nyand  diners  naturall  queftions:  asfpeciallyfin  Aqueducdes .  For  in  their 
courfes,  leadinges  about,  in  the  leuell ground,  and  in  the  mountinges ,  the  natu * 
rail  Spirit  es  or  breathes  areingendred  diners  Wayes  :  The  hindrances ,  which 
they  caufe,  no  man  can  helpe,  but  be, 'Which  out  ofThilofophie,  hath  learned  the 
originall  caufes  of thinges .  Like  wife , who foeuer  flail  read  Ctefibius,  or  Ar» 
chimedes  bookesf  and  of others, who  haue  written fuel?  flules)can  not  thinke,as 
they  do :  'bnleffe  he  f  ull  haue  receaued  of  Thilofophers ,  inflruCiions  in  thefe 
thinges  .  And  Mufike  he  mufti  nedes  know :  that  he  may  haue  lender  standing, 
both  of  Tegular  and  Mathematicall  Mufike:  that  he  may  temper  Well  his  Ba* 
liftes,  Catapultes , and  Scorpions.  iyc.  Moreouer ,the  Brafen  Veffels , which  in 
T  beatres,ate  placed  by  Mathematicall  order, in  ambries  fender  the  Steppes:  and 
the  diuerfities  of  the foundes  (which y  Grecians  call  w  )  are  ordred  according 
to  Mu  f  call  Symphonies  W  Harmonies:being diflributed in y  Circuites, by  (Di* 
ateffaronJDiapente,and  (Diapafon.  That  the  conuenient  boy  ce,  of  the  players 
found, whe  it  came  to  thefe  preparations,  made  in  order ,  there  being increafed: 
Withy  increafing, might  come  more  cleare  &  pleafant,to y  eares  of  the  lokers  on. 
i&c.And  ofAJlronomieJs  knoweyEaSt, Weft, Southland  North .  The  fajhion 
of  the  heauen ,  the  AEquinox ,  the  Solfticie ,  and  the  courfe  of  the fibres.  Which 
thinges grnleafl  one  know:he  can  not perceiue ,any  thyng  at  all, the  reafon  of  Ho* 
rologies.Seyng  ther fore  this  ample  Science ,is  garnified ,  beautified  and  flored. 
With  fo  many  and  fundry  f  kils  and  knowledges:!  thinke , that  none  can  iuflly  ac* 
count  themfidues  ArchiteCfes ,of  the  fnddeyne.  But  they  onely,who  from  their 
childes yeares ,afcendyng  by  thefe  degrees  of  knowledges,  beyngfoUered  bp  with 
the  atteynyng  of  many  Languages  and  Artes  ,  haue  Wonne  to  the  high  T aber* 
nacle  of  Archiffiure.is'c.And  to  whom  Nature  hath giuen fuch  quicke  Circum* 
fieCfionjbarpnes  of  Witt,  and Memorie,that  they  may  be  bery  abfoktelyfkilU 
fullin  Geometrie , . Afironomie ,  Mufike,  and  the  reft  of  the  Artes  Mathemati * 


John  Dee  his  MatliematicallPrrface, 

calkSticb  fur mount  and pajfe  the  callyng3andftatt',  of  Ar  chit e  tides:  and  are  he*  A  t&fatbe* 
come  Mathematiciens.e&c.Mndthey  are found feldome.  As,  in  tymes  paft  } 1 \>as  matieien, 
Mriftarchus*Samius:Tbildlaus3andArcbytasfTarentynes:  Apollonius  Bergpus: 
Eratofthenes  Cyreneus: Archimedes 3and  SeopasJSyracufians.  Who  alfojeft  to 
theyr pofteritiepnany  Engines  andGnomomcaUypork.es:  by  numbers  andnatu* 
rail  meanes  ftnuented  and  declared. 

Thus  much.,  and  the  fame  wordes  (in  fenfejin  one  onely  Chapter  of  this  Inco¬ 
parable  ArchitectVitruutus,{CdX\  youfinde.And  if you  (hould  ,  hut  take  his  boke-in 
your  hand,and  (lightly  loke  thorough  it, you  would  fay  ftraight  way :  This  is  Geo-  yitrmws* 
metrie,Anthmetike, ^Astronomic, Mufike,Anthropographie,fJjdragogie,  Horometrie.&c, 
and  (to  coclude )  the  Storehoufe  of  all  workmafhip .  No  w,let  vs  liften  to  our  other 
Iudge,our  FlorcntmeyLeo.Baptifta:a.nd  narrowly  confider,how  he  doth  determine 
of  Architecture.  Sed  anteoy  vltra  progrediar.&c.  But  before  Iprocede  any  further 
((ayth  he)/  thinkefthat  I  ought  to  exprejfe  7  yohat  man  I  leould  haue  to  bee  al* 
lowed  an  Architect.  For  ft  ypill  not  bryng  in  place  a  Carpenter :  as  thoughyou 

might  font. pare  him  to  the  Chief  M afters  of other  jtirtes.  For  the  hand  of the 
•  farpenterfs  the  Ar  chit  e  tides  Inftrument,  But  I  ypill  appoint  the  Architect  to  be 
that  manfttoho  hath  the J kill  ft  by  a  certaine  and  meruailous  meanes  and  1 Vayft  n  tu  ' 
both,  in  mindeand  Imagination  to  determine:  and alfo  in  yporke  to  finifh :  yphat  ” 
yoorkes foeuerfty  motion  of ypaight}and  cupplingand  framyng  together  of  bo* 
dyespnay  moH  aptly  be  Qjmtnodious for  the  yportbieft  Vjes  of  Man,  And  that  he  » 
may  be  able  toperforme  thefe  thinges }  he  hath  nede  of  atteynyng  and  knowledge 
of  the  beft3and  moft  yporthy  thynges.  tyre .  1  he  yphole  Feate  of Architecture  in 
buildy ngponfUeth  in  Line  ament es  3and  in  Framyng .  ylna  the  yphole  power; 

and fktll  ofLineamentesftendetb  to  this :  that  the  right  and  ahfolute  Ip  ay  may 
he  had3of Coaptyngand  ioyning  Lines  and  angles'.by  yohich  3the face  of  the  buiti 
dyng  or  frame  pnay  be  comprehended  and  concluded ,  And  it  is  the  property  of 
Lineamentes  pto  prefcribelmto  buildynges  }and  euery  part  of tbem3an  apt  place , 

CT  certaine  ntiber :  a  Worthy  maner}and  afemely  order :  thatfoft  yphole  forme 

and  figure  of the  buildyngpnay  reft  in  the  1?ery  Lineament  es. (type .  And  ype  may  *  The  Im- 

preferibe  inmynde  and  imagination  the  ypbole  formes  ft  all  materiall ftuffe  be*  matcrialitl" 

Jtig  ft  eluded Which  point  ype f hall  atteynefty  Fbotyng  and  forepointyng  the  an* 

glespnd  lines  fty  a  Jure  and  certaine  diretiiion  and  connexion.  Seyng  thenfthefe 

thinges  y  are  thus :  Lineamente 3fhalbe  the  certaine  and  constant  hrefcribyng3  WbatftAnu- 

concerned  in  mynde:  made  m  lines  and  angks:and finiftbed  ypith  a  learned  minde  mntts' ' 

andypyt.  We  thanke  you  Maher  Baptist,  that  you  haue  fo  aptly  brought  your  „ 

Arte ,  and  phrafe  therof ,  to  haue  (ome  Mathematical!  perfection :  by  certaine  or-  ,,  <3S(jte, 
der,  nuber,  forme,  figure,  and  Symmetric  mentall:  all  naturall  &  fenfible  ftuffe  fet  a  » 
part.  N ow, then, it  is  euidcnt,(Gentle  reader)how  aptely  and  worthely ,  I  haue 
preferred  Architecture,  to  be  bred  andfoftered  vp  in  the  Dominion  of  the  pereles 
Princefe,  (JUathematicx :  and  to  be  a  naturall  SubieCt  of  hers .  And  the  name  of 
Architecture,  is  of  the  principalitie, which  this  Science  hath,  aboue  all  other  Artes. 

And  Plato  affirmeth  ,  the  Architect  to  be  AH  after  ouer  all, that  make  any  worke. 
Wherupon,he  is  neither  Smith,nor  Builder:  nor3(eparatcly,  any  Artificer:  but  the 

d.iiij.  Hed, 


Anno.  IJ5P, 


lohnDee  his  Mathematical!  Preface. 

Hed,thcProuoft,  the  Dirc£ter,and  Iudge  of  all  Artificial!  workes,  and  all  Artifi- 
cers.For,thc  true  Architects is  hable  to  teach ,D em  onft  rate,diftribu  te,dcfcrib c ,  and 
Iudge  all  workes  wrought.  And  he,onely,Iearcheth  out  the  caufes  and  reafons  of 
all  Artificial!  thynges.Thus  excellent, is  Architecture : tho ugh  few(in  our  dayes)at- 
teyne  thereto  :  yet  may  not  the  Arte, be  otherwife  thought  on,  then  in  veiy  dede 
it  is  worthy. Nor  we  may  not, of  auncient  Artes,make  new  and  impeded  Definiti¬ 
ons  in  our  dayesrfor  fcarfitie  of  Artificers  :  No  more, than  we  may  pynche  in, the 
Definitions  of  Wifedomc, or  Honefite ,  or  of  Frendejhyp  or  of  Iujlice .  No  more  will 
I  confent,to  Diminifli  any  whit, of  the  perfedionand  dignitie ,  ("by  iuft  caufe  )  al¬ 
lowed  to  abfolute  Architecture.  V  nder  the  Diredion  of  this  Arte ,  are  thre  prin- 
cipall,necdTary  Mechanic  all  Artes .  Namely ,  Horvfwg ,  Fortification ,  and  T{aupegie. 
Horvfingj  I  vnderftand,both  for  Diuine  Seruice,and  Mans  common  vfage: publike, 
and  priuate.Of  Fortification  an dNaupegie,  ftraunge  matter  might  be  told  you:  But 
percnaunce/ome  will  be  tyred, with  this  Bederoll,  all  ready  rehearfcd:  and  other 
fomc,  will  nycely  nip  my  grofle  and  homely  difcourfing  with  you  :  made  in  poll: 
haft :  for  fearc  you  fliould  wante  this  true  and  frcndly  warny ng,  and  taft  giuyng, 
of  the  Power  JSIathematicall .  Lyfe  is  ihort,  and  vncertaine  :  Tymes  are  periloufe; 
Bcc .  And  ftill  the  Printer  awayting,  for  my  pen  ftaying :  All  thefe  thingcs,with 
farder  matter  of  I-ngratefulnes,  giue  me  occafion  to  paffe  away ,  to  the  other  Artes 
remainyng,  with  all  fpede  pofsible. 

Xhc  Arte  of  Nauigation,  demoiiftrateth  how, by  the  fhorteft 
good  way,  by  the  apteft  Diredtio,&  in  the  fhorteft  time,  a  fufficient 
Ship,betwene  any  two  places  (in  paffage  Nauigable,) afsigned :  may 
be  codinded:  and  in  all  ftormes,&  naturall  difturbances  chauncyng, 
how, to  vie  the  beft  pofsible  meanes ,  whereby  to  recouer  the  place 
fir  ft  afsigned .  What  nede ,  the  Mafler  Pilote ,  hath  of  other  Artes ,  here  before 
recited, it  is  eafie  to  knowras,  of  Hydrographies  AChonomky^ACirologie ,  and  Horome- 

trie .  Prefuppofing  continually, the  common  Bafe,andfoundacion  or  all:  namely 

Arithmetike  and  Geometric.  So  that,he  be  hable  to  vndcrftand,and  Iudge  his  own 
neceflary  Inftrumehtes,and  furniture  N eceftary:  Whether  they  be  pcrfedly  made 
or  norand  alfo  can, (if nede  be) make  them,hym  fclfe.  As  Quadrantes,  The  Aftro- 
nomers  Ryng,The  Aftronomers  ftaffe,The  Aftrolabc  vniucrfaii.  An  Hydrogra- 
phicall  Globe.Charts  Hydrographicall,true,(not with  parallell  Meridians) .  The 
Common  Sea  CompasrThe  Compas  of  variacion:  The  Proportionalfand  Para- 
doxall  Compafle$(of  melnuented,forourtwo  Mofcouy  Mafter  Pilotes,atthc  re- 
queft  of  the  Company)  Clockes  withfpryng:  houre,halfe  houre,and  three  houre 
Sandglalfes: & fundry  other  Inftrumetes:  And  alfo,  be  hable,on  Globe,  or  Playne 
todeferibe  the  Paradoxall  Compalfe :  andduely  to  vie  the  fame,to  allmanerof 
purpofes,  whereto  it  was  inuented.  And  alfo,  be  hable  to  Calculate  the  Planetes 

places  for  all  tymes.  ,c  .  T 

Moreouer,with  Sonne  Monc  orSterrefor  without)be  hable  to  define  the  Lon- 
gitude  Be  Latitude  of  the  place, which  he  is  in:  So  that,the  Longitude  &  Latitude 
of  the  place,from  which  he  fayled,be  giuen :  or  by  him, be  knowne.  whereto, apper- 
tayneth  expert  meanes, to  be  certified  euer,of  the  Ships  way .  &c.  And  by  toreie- 
ing  the  Rifing,Settyng ,  N oneftedyng ,  or  Midnightyng  of  cer taine  tempeftuous 
fixed  Sterres  :  or  their  Coniun<ftions ,  and  Anglynges  with  the  Pranetes ,  &c.he 
ought  to  haue  expert  coniedure  of  Stormes ,  T empeftes ,  and  Spoutes :  and  iuch 
lyke  Meteorological!  effe<ftes,daungerous  on  Sea.  For  (as  Plato  fayth  y)Mutanonest 


lohn  Dee  his  Mathematical!  Preface. 

epportunitatefy  tempo  rum  prefentire,  non  minus  rei  militari,  quam  Agriculture?  Nauiga - 
tionitfc  conuenit.  T o  forefee  the  alterations  and  opportunities  of  tymesjs  come * 
nient ,  no  lefe  to  the  Art  of  JVarre }  then  to  HuJ  bandry  and  Navigation.  And 
befides  fuch  cunnyng  meanes ,  more  euident  tokens  in  Sonne  and  Mone ,  ought 
of  hym  to  be  knowen:  fuch  as(the  Philofophicall  Po ex€)Virgilius  teacheth,  in  hys 
GeorgikesAN'ncxc  he  fayth, 

Sol  quofy  dr  exoriens  dr  quum  fe  condet  in  vndas, 

Signa  dabitySolem  certifima  ftgna fequuntur.drc . 

- - —  'Horn  ftpe  videmns , 

Ipjtus  in  vultu  varios  err  are  colores. 

Caruleus,  pluuiam  denunciat  dgneus  Euros. 

Sin  macule  incipient  rutilo  immifcerier  ign  ’t , 

Omnia  turn  par  iter  vento^nimbifq,  videbis 
Eeruere:  non  ilia  quifquam  me  nocie  per  altum 
Ire ,  necp  a  terra  moueat  conuellere  funem .  dfc. 

Sol  tibi ftgna  dabit. Salem  quis  dicerefalfum 
lAudeati - - drc. 

And  fo  of  Mone,  Sterres, Water, Ayre, Fire, Wood, Stones, Birdes, and  Bcaftes, 
and  of  many  thynges  els,a  certaine  Sympathicall  forewarnyng  may  be  had:  fome- 
tymes  to  great  pleafure  and  proflit ,  both  on  Sea  and  Land.  Sufnciendy,  for  my 
prefent  purpofe ,  it  doth  appeare,  by  the  premiffes ,  how  Mathematically  the  Arte  of 
Navigation,  is:and  how  it  nedeth  and  alfo  vfeth  other  Mathematicall  Artes  :  And 
now,  if  I  would  go  about  to  fpeake  of  the  manifold  Commodities,  commyng  to 
this  Land,  and  others,  by  Shypps  and  Navigation ,  you  might  thinke ,  that  I  catch 
at  occafions ,  to  vfe  many  wordes ,  where  no  nedc  is. 

Y et,  this  one  thyng  may  I,  (iuftly)  fay.  In  Nauigaiion, none  ought  to  haue  grea¬ 
ter  care, to  be  fkillfull,then  ourEngiiih  Pylotes.  And  perchaunce,Some,  would 
more  attempt:  And  other  Some,more  willingly  would  be  aydyng,  if  they  wift  cer- 
tainely,What  Priuiledge,God  had  endued  this  Hand  wirh,by  reafon  of  Situation, 
moft  commodious  for  Nanigation,  to  Places  moft  Famous  8c  Riche.  And  though, 
(of*  Late)  a  young  Gentleman,a  Courragious  Capitaine ,  was  in  a  great  ready- 
nes, with  good  hope,  and  great  caufes  ofperfuafion,to  haue  ventured,  for  a  Dll- 
CO  lierye,  (either  Wejlerly,  by  Cape  de  Paramantia :  or  ,  aboue  Noua  Zemla, 
and  the  Cjremijfes)and  was, at  the  very  nere  tyme  of  Attemptyng  ,  called  and  em¬ 
ployed  otherwife(both  then, and  fince,)in  great  good  feruice  to  his  Countrey ,  as 
the  Irifh  Rebels  haue  *  tailed:  Yet, I  fay,  ( though  the  fame  Gentleman ,  doo  not 
hereafter, deale  therewith)Some  one, or  other ,fhould  liften  to  the  Matter:  and  by 
good  aduife,and  diferete  Circumfpe&ion ,  by  little,  and  little,  wynne  to  the  fuffn 
cient  knowledge  ofthatTradeand Voyage: Which,  now,  I  would  be  Tory, 
(through  CarelefnefTe,want  of  Skill, and  Courtage, )  fhould  remayne  Vnknowne 
and  vnheard  of.  Seyng,  alfo,we  are  herein,  halfe  Challenged,  by  the  learned,by 
halfe  requeft,publifhed.  Therof,verely,  might  grow  Commoditye ,  to  this  Land 
chiefly,  and  to  the  reft  of  the  Chriften  Common  wealth,  farre  pafling  all  riches 
and  worldly  Threafure. 

Thaumaturglfee,is  that  Art  Mathematicall,  which  giueth  cer¬ 
taine  order  to  makeftraunge  workes ,  of  the  fenfe  to  be  perceiued, 

and  of  men  greatly  to  be  wondred  at.  By  fundiy  meanes,this  Wonder- 
worke is  wrought.  Some,by  Fneumatithmie .  As  the  workes  of  Ctefibius  and  Hero , 

A.j.  Some 


Georgia,  i. 


*Anno.\%6j 

s.H.g . 


*Anno.i$6g 


lohtt  Dee  liis  Mathematical!  Preface. 

Soineby  waight.wherof T imam  lpeaketh.Some,by  Strirtges  ftrayned,or  Springs, 
therwith  Imitating  liucly  Motions.Some,  by  other  meanes,as  the  Images  of  Mer¬ 
curic  .-and  the  brafen  hed,fnade  by  Albertus  Magnus,  which  dyd  feme  tp  Ipeake.-Btn?- 
thins  was  excellent  in  thefe  feates.  T  o  w  h  o  m  ,CaJsi  od or  us  writyng,fayth.2  our  put* 
pofe  is  to  know  profound  thyngesiand  to jkew  meruajles.  <By  the  diSpofition  of 
jour  Arte ,  Metals  do  low :  (Diomedes  ofbrafie ,  doth  blow  a  Trumpet  loude  :  a 
brafen  Serpent  hiffetkbyrdes  made  ,fingfwetelj.  Small  thynges  lee  rehearfe 
ofyou,loho  can  Imitate  the  heauen.ejrc.  Of  the  ftraunge  Selfmouyng,  which,  at 
*4mo.  1551  Saint  Denys ,  by  Paris  ,*Ifaw ,  ones  or  twife  (  Orontius  beyng  then  with  me,  in 
C ompany  jit  were  to  ftraunge  to  tell.  But  fome  haue  written  it.  And  yet,  (I  hope) 
it  is  there, of  other  to  be  fene.  And  by  PerJpecJiue  alio  ftraunge  thingcs,arc  done.  As 
partly  (before)  I  gaue  you  to  vnderftandin  Perjpetfm.  As,  to  fee  in  the  Ayre,  a  loft, 
thelyuely  Image  of  an  other  man ,  either  walkyng  to  and  fro  :  or  ftandyng  ftill. 
Likewife,  to  come  into  anhoufe,and  there  to  fee  the  liucly  ftiewofGold,Siluer 
or  precious  ftones:and  commyng  to  take  them  in  your  hand ,  to  finde  nought  but 
Ayre. Hereby,  haue  fome  men  ( in  all  other  matters  counted  Wife  )  fouly  ouerfnot 
the  fellies :  mifdeaming  of  the  meanes.Thaforeiayd  CUudmCAeflinus.  Hodie  mag- 
na  literature  viros  &  magne  refutdtioms  ‘videmus ,  opera  qua  dam  quafi  miranda  ,fupra 
Nat ura  putare:  de  qmbnsin  PerjfeBiua  dobhis  caufamfdciliter  reddidijfit.Thzt  is  JSlow 
a  dayes  ,!»e fee  fome  men, yea  of  great  learnyng  and  reputation,  to  Iudge  certain 
1 V0rk.es  as  meruaylous  ,ahone  the  power  of  Mature :  Of lefich  1 vorkes,one  that 
leerefkillfull  in  Terfpeltiue  might  eafely  hauegiuen  the  faufe.  Of  Archimedes 
Sphere, Cicero  witneftqth .Which  is  very  ftraunge  to  tftinke  on.  For  when  Archie 
medes{£ ayth  he) did  faflen  in  aSph  cere , the  mouynges  of  the  Sonne,  Mone,and of 
the  fine  other  r?lanets,he  did, as  the  God,H>hiclfin  Timeeus  of?  Into)  did  make 
the  leorld.  That  ^one  turnytig,Jhould  rule  motions  mo  filmlike  in flownes,  arid 

fw  fines.  But  a  greater  caufe  of  meruayling  we  haue  by  CUudianm  report  hereof. 
Who  aiftrmeth  this  Archimedes  worfo,  to  haue  ben  of  Glafte.  And  difeourfeth  of  it 
more  at  large:  which  I  omit.  The  Doue  ofwood ,  which  the  Mathematicien  Ar¬ 
ch  jt  as  did  make  to  fly e, is  by  iMgellius  ipoken  of.O  tDadalus  ftraunge  Images,  Plato 
reporteth .  Homer  e  of  Vulcans  Selfmowtrs ,  (by  fccret  whcles)lcaueth  in  writyng .  Ari- 
fiotkja  hys  Volmkes,  of  both,  makethmention.  Meruaylous  was  the  workeman- 
-£ftyp,of  late  dayes, performed  by  good f  kill  otTrochilike.  dec  •  Tor  in  Noremberge, 
A  flye  oflern, beyng  let  out  of  the  Artificers  hand,did(as  it  were)fly  about  by  the 
geftes,at  the  table, and  at  length, as  though  it  were  weary ,  rctoutne  to  his  mafters 
hand  agayne .  Moreouer,  an  Artificial!  Egle ,  was  ordred ,  to  fly  out  of  the  fame 
Towne,a  mighty  way, and  that  a  loft  in  the  Ayre,  toward  the  Empcroiir  comming 
thethenand  followed  hym,  beyng  come  to  the  gate  of  the  townc.*  Thus,you  fee, 
What,ArteMathematicallcan perform e,when Sldll ,  will,  Induftry,  andHabili- 
ty,  are  duely  apply ed  to  profe. 

A  DC  ef'o't  An  d  for  thefe, and  fuch  like  marueilous  Ades  and  Feates, Naturally  ,Mathc- 

ApologeJca/L  matically,and  Mechanically,  wrought  and  contriued :  ought  any  honeft  Student, 
and  Modeft  Chriftian  Philofopher,be  countcd,&  called  a  Coniurer  ?  Shall  the 
folly  of  Idiotes,  and  the  Mallice  of  the  Scornful!,  to  much  preualle,  that  He,  who 
feeketh  no  worldly  game  or  glory  at  their  handes  :  But  onely,of  God, the  threafor 
of  heauenly  wifedome,&  knowledge  of  pure  veritie :  Shall  he  (I  fay)  in  the  meane 


SDe  his  qua 
Munch  mi- 
rabiltter  eue- 
niunt.  cap.  8 


Thfc.  i. 


ST 


lohii  Dee  his  Mathematical!  Preface* 

fpace,  be  robbed  and  fpoiled  of  his  honed:  name  and  fame  <  He  that  feketh  (  by  Si 
Paules  aduertifement )  in  the  Creatures  Properties ,  and  wonderful!  vertues,  to 
finde  iufte  caufc,  to  glorifie  the  Asternalfand  Almightie  Creator  by :  Shall  that 
man,  be  ( in  hugger  mugger  )  condemned,  as  a  Companion  of  the  Helhoundes , 
and  a  Galley  and  Coniurer  of  wicked  and  damned  Spiritesc’  He  that  bewaileth  his 
great  want  of  time,fufficient(to  his  contentation)for  learning  of  Godly  wifdome, 
and  Godly  Verities  in  :  and  onely  therin  fetteth  all  his  delight :  Will  that  ma  leefe 
and  abufe  his  time,  in  dealing  with  the  Ghiefe  enemie  of  Chrift  our  Redemer:  the 
deadly  foe  of  all  mankinde  :  the  fubtile  and  impudent  peruerter  of  Godly  Veritie: 
the  Hypocriticall  Crocodile:  the  Enuious  Bafilifke,  continually  defirous,  in  the 
twinkeofaneye,todeftroy  all  Mankinde,  both  in  Body  and  Soule  ,  eternally  i  . 

Surely  (for  my  part, fomewhat  to  fay  herein)  I  haue  not  learned  to  makefobrutifh, 
and  fo  wicked  a  Bargaine .  Should'  I,  for  my  xx.or  xxv.  yeares  Studie  :  for  two  or 
three  thoufand  Markes  fpending :  feuen  or  eight  thoufand  Miles  going  and- trauai- 
ling, onely  for  good  learninges  lake  r  And  that,  in  all  maner  of  wethers  :  in  all  ma- 
ner  of  waies  and  paftages  :  both  early  and  late :  in  daunger  ofviolence  by  man :  in 
daunger  o.fdcftru&ion  by  wilde  beaftes :  in  hunger :  in  third: :  in  perilous  heates 
by  day,  with  toyle  on  foote :  in  daungerous  dampes  of  colde,by  night,  almoft  be- 
reuing  life  :  (as  God  knoweth) :  with  iodginges,  oft  times,to  iinall  eafe  :  and  font-  , 
time  to  Idle  fecuritie.  And  for  much  more  (then  all  this)  done  &  fuifred,  for  Lear¬ 
ning  and  attaining  of  Wifedome :  Should  I  ( I  pray  you)  for  all  this,no  otherwife, 
nor  more  warily :  or  (by  Gods  mercifulnes  )  no  more  luckily,  haue  fiftied,  with  fo 
large, and  coftly,a  N ette,  fo  long  time  in  drawing  (and  that  with  the  helpe  and  ad- 
uife  of  Lady  Philofophie,&  Queen  e  Theologic)  :  but  at  length,  to  haue  catched, 
and  drawen  vp,*  a  Frog  i  Nay, a  Deuill  <  For,fo,doth  the  Common  peuifh  Pratler  *  ^ prouerfa 
Imagine  and  Iangle:  And,fo,doth  the  Malicious  fkorner,fecretly  wiftie,&  brauely  FaJfefi[bt, 
and  boldly  face  down, behinde  my  backe .  Ah, what  a  miserable  thing, is  this  kinde  ^caught 
of  Men  <  How  great  is  the  blindnes  &  boldnes,of  the  Multitude, in  thinges  aboue  re&% 
their  Capacitie  <  What  a  Land:  whata  People :  what  Man  ers :  what  Times  are 
thefe  <  Are  they  become  Deuils,them  felues:  and, by  falfe  witnelfe  bearing againft 
their  Neighbour,  would  they  alfo,  become  Murderers  *  Doth  God, fo  long  geue- 
them  refpite,  to  reclaime  them  felues  in,  from  this  horrible  {laundering  of  the  gilt- 
leife :  contrary  to  their  owne  Confciences :  and  yet  will  they  notceafe  <  Doth  the 
Innocent,forbeare  the  calling  of  them,  Juridically  to  aunfwerehim,accordingto 
the  rigour  of  the  Lawes  :  and  will  they  defpife  his  Charitable  pacience  1  As  they, 
aeainft  him,  by  name,  do  forge, fable, rage, and  raife  {launder,  by  Worde  &  Print: 

Will  they  prouoke  him,  by  worde  and  Print, likewife,  to  N ote  their  Names  to  the 
World :  with  their  particular  deuifes, fables, beaftly  Imaginations,  andvnchriften- 
like  {launders  c’  Well :  Well .  O  (you  fuch  )  my  vnkinde  Countrey  men  .  O  vn- 
naturall  Countrey  men  .  O  vnthankfull  Countrey  men  .  O  Brainficke,  Raflie, 
Spitefull,and  Difdainfull  Coun trey  men  .  Why  opprefteyou  me,  thus  violently, 
with  yo  ur  {laundering  of  me :  Contrary  to  Veritie:  and  contrary  to  your  owne 
Confciences  <  And  I,  to  this  hower,  neither  by  worde,  deede,  or  thought,  haue 
ben e, any  wav, hurtfull, damageable, or  iniurious  to  you,or  yours  <  Haue  I,fo  long, 
fo  dearly, fo  farre,fo  carefully, fo  painfully ,fo  daungcroufly  fought  6c  trauailed  for 
fhelearnlng  of  Wifedome,&  atteyning  ofVertue  :  And  in  the  end(ln  youriudge- 
met)am  I  become, worfe, then  when  I  begaf  Worfe,the  a  Mad  man.?  A  dangerous 
Memberin  the  Common  Wealth:  and  no  Member  of  the  Church  of  Chrift?  Call 
you  this, to  be  Learned  ?  Call  you  this, to  be  a  Philofopher  ?  and  a  louer  of  Wife- 
dome  ?  To  forfake  the  ftraight  heauenly  Way  *.  and  to  wallow  in  the  broad  way  of 

A.ij.  dam- 


Iohn  Dee  his  Mathematicall  Preface . 

damnation  ?  To  forfake  the  light  of  heauenly  Wifedome:  and  to  luike  in  the  dun¬ 
geon  of  thePrinee  of  darken  ene  ?  To  forfake  the  Veritie  of  God,&his  Creatures: 
and  tofawne  vpon  the  Impudent, Craftic,Obftinate  Tier,  and  continuall  difgracer 
of  Gods  V  eritie,  to  the  vttermoft  of  his  power  l  T  o  forfake  the  Life  &  Bliffe  Aiter- 
nall:  andto  cieauevntothe  Author  of  Death  euerlafting  ?  that  Murderous  Tv- 
rant,  moft  gredily  awaiting  the  Pray  of  Mans  Soule  ?  Well  :  I  thanke  God  and 
ourLordelefus  Chrift,  for  the  Comfort  which  I  haue  by  the  Examples  of  other 
men,  before  my  time :  To  whom, neither  in  godlines  of  life,  nor  in  perfection  of 
learning,  I  am  worthy  to  be  compared :  and  yet,  they  fuftained  the  very  like  Iniu- 
ries ,  that  I  do  :  or  rather, greater.  Pacient  Socrates ,  his  ^Apologie  will  teflifie :  Apu- 
leius  his  ^Apologies,  will  declare  the  Brutifhneffe  ofthe  Multitude  .  loannes  Via*. r, 
Earle  of  Mirandula,  his  Apokgie  will  teach  you,  of  the  Raging  flaunder  ofthe  Ma¬ 
licious  Ignorant  againft  him  .  Joannes  T rithemim,  his  Apologie  will  fpecifie,  how 
he  had  occafion  to  make  publike  Proteftation :  as  well  by  reafon  of  the  Rude  Sim¬ 
ple  :  as  alfo,in  refpeft  of  fuch,as  were  counted  to  be  of  the  wifeftfort  of  men.  Ma- 
„  ny  could  I  recite  :  But  I  deferre  the  precife  and  determined  handling  of  this  mat- 
O**  „  ter:  being  loth  to  deted  the  Folly  &  Mallice  of  my  N  atiue  Countrey  men.*Who, 
„  fo  hardly,  can  difgeft  or  like  any  extraordinary  courfe  of  Philofophicall  Studies: 
,,  not  falling  within  the  Cumpaffe  of their  Capacitie :  or  where  they  are  not  made 
priuie  of  the  true  and  fecrete  caufe,  of  fuch  wonderfull  Philofophicall  Feates. 
Thefe  men,  are  of fower  fortes,  chiefly .  The  firft,  I  may  name,  Faine pratling  bu- 
fte  bodies  .-The  fecond ,  Fond  Frendes :  The  third,  Imperfectly  serious:  and  the  fourth, 
t Malicious  Ignorant  .  To  cche  of  thefe  (briefly, and  in  charitie  )  I  will  fay  a  word 
I#  ortwo,andforeturne  to  my  Preface.  Vaine  pratling  bufe  bodies ,vfe  your  idle 
allemblies,and  conferences,  otherwife,  then  in  talke  of  matter,  either  aboueyour 
Capacities,  for  hardnefle :  or  contrary  to  your  Confciences,  in  Veritie  .  Fonde 
2 .  Frendes,  leaue  of,fo  to  commend  your  vnacquainted  frend,vpon  blindeaTedion: 
As,  becaufe  he  knoweth  more, then  the  common  Student:  that,  therfore,  he  inuft 
needes  be  fkiifull,andadoer,infuch  matterandmaner,asyouterme  Coniuring. 
Wecning,thereby,  you  aduaunce  his  feme :  and  that  you  make  other  men,  great 
marueilers  of  your  hap,  to  haue  fuch  a  learned  frend  .  Ceafe  to  aferibe  Impietie, 
where  you  pretend  Amitie .  For, if  your  tounges  were  true,  then  were  thatyour 
frend,  Vntrue,  both  to  God, and  his  Soucraigne .  Such  Frendes  and  Fondlinges,  I 
fliake  of,  and  renounceyou :  Shakeyou  of,  your  Folly.  Imperfectly  serious, to  you, 
3*  do  I  fey:  that  (perhaps)  well,  do  you  Meane :  But  farre  you  mifle  the  Marke :  If  a 
Lambe  you  will  kill,tofeede  the  flocke  with  his  bloud.  Sheepe,  with  Lambes 
bloud,  haue  no  naturall  fuftenaunce :  No  more,  is  Chriftes  flocke,  with  horrible 
flaunders,  duely  edified.  Nor  your  feire  pretenfe,  byfuch  rafhe  ragged  Rheto- 
rike,any  whit,well  graced.  But  fuch,asfo  vfe me, will  finde  a  fowle  Gracke  in  their 
Credite .  Speake  that  you  know  :  And  know,  as  you  ought :  Know  not, by  Heare 
Eiy,  when  life  lieth  in  daunger.  Search  to  the  quicke,&  let  Charitie  be  your  guide. 
4.  c Malicious  Ignorant ,  what  (hall  I  fey  to  thee  ?  Prohibe  linguam  tuam  a  malo .  ^Ade - 
tr action  c  parcite  lingua: .  Caufe  thy  toung  to  refraine  fro  euill.  fefraineyour  toung 
from  flaunder .  Though  your  tounges  be  fliarpned.  Serpent  like,  &  Adders  poy- 
Pfai,  1 40 .  fon  lye  in  your  lippes  :  yet  take  heede,and  thinke, betimes,  with  your  felfe.  Fir  lin¬ 

gua  fm  non  flabilietur  in  terra .  Firum  violentum  venabitur  malum ,  donee pracipitetur. 
For,fure  I  am,  fhtja  faciet  Dominus  Iudicium  afflicti  .•  &  vindictam pauperum . 

Thus,  I  require  you,  my  aflured  frendes,  and  Countrey  men  (  you  Mathemati- 
dens,Mechaniciens,andPhilofophers,Charitable  and  diferete)  to  deale  in  my 

behalfe. 


lohn  Dee  his  Matliematicail  Preface. 

behalf, with  the  light  &  vntrue  toungcd,  my  enuious  Aduerfaries,or  Fond  fr ends » 

And  farther,  I  would  wifhc,  that  at  leyfor,  you  would  confider,how  Bafdius  Mag* 
nus,  layeth  CMofes  and  Daniel:,  before  the  eyes  of  thofe,  which  count  all  fuch  Stu¬ 
dies  Philofophicall  (as  mine  hath  bene)  to  be  vngodly ,  or  vnprofitable .  Waye 
well  S. Stephen  his  witnefle  of  CMofes .  Erudttus  est  CMofes  omni  Sapientta  JUgyptioru:  ^  £ 

&  eratpotens  in  verbis  &  openbm fuis .  Mofes  leas  inRmfifed  in  all  rnaner  of^ife* 
dome  of  the  ^Egyptians :  and  he  "ivas  of  power  both  in  his  hordes ?  and  forties. 

You  lee  this  Philofophicall  Power  &  Wifcdome, which  CMofes  had, to  be  nothing 
mifliked  of  the  Holy  Ghoft.  Yet  Plinius  hath  recorded,  Mofes  to  be  a  wicked  Magi* 
cien .  And  that  (of  force)  muft  be,  either  for  this  Philofophicall  wifedome,iearned, 
before  his  calling  to  the  leading  of  the  Children  of  Jfrael :  or  for  thofe  his  won- 
ders,wrought  before  King  Pharao ,  after  he  had  the  conducing  of  the  Ifraelites.  As 
concerning  the  frft,you  perceaue,  how  S. Stephen,  at  his  Martyrdome  (  being  full 
of  the  Holy  Ghoft)  in  his  Recapitulation  of  the  olde  Teftament,hath  made  men¬ 
tion  of  Mofes  Philofophie :  with  good  liking  of  it :  And  Baflius  Magnus  alfo,  auou- 
cheth  it,  to  haue  bene  to  Mofes  profitable  (  and  therefore,  I  fay,  to  the  Church  of 
God,  neceflary).  But  as  cocerning  Mofes  wonders, done  before  King  PharaovGod , 
him  felfe,  fayd  :  Vide  vt  omnia  osfenta,  qua  pofui  in  manu  tua ,  facias  coram  Pharaowe. 

See  that  thou  do  all  thofe  bonders  before  Pharao  ?  Tvhicb  I  haue  put  in  thy  hand. 

Thus, you euidently perceaue, how vaShly^P  limits hath llaundered  Mofes, ofvayne  Lib. 30. 
fraudulent  CMagike,  faying  :  EH  Sr  alia  Magices  F  actio,  a  CMofe ,  Iamne,&  Iotape,  Iii*  Cap,  1. 
dais  pendens :  fed  multis  millibus  annorum pojl  Zoroastrem.frc.  Let  all  ftich ,  there¬ 

fore,  who,  in  Iudgement  and  Skill  of  Philofophie,  are  farre  Inferior  to  Plinie,  take 
good  heede,leaft  they  ouerfhoote  them  felucs  rafhly ,  in  Iudging  of  Phtlofophcrs 
Hraunge  Actes  •  andtheMeanes,how  they  are  done .  But,  much  more, ought  they 
to  beware  of  forging,  deuifing,  and  imagining  monftrous  feates,  and  wonderfull 
Workes,  when  and  where,  no  fuch  were  done :  no,  not  any  fparke  or  likelihode,of 
fuch, as  they, without  all  fhame,  do  report .  And  ( to  conclude  )  moft  of  all,  let 
thembea!hamedofMan,andafraide  of  the  dreadfullandlufteludge:  bothFo- 
iiflily  or  Malicioufly  to  deuife :  and  then,deuililhly  to  father  their  new  fond  Mon- 
fters  on  me :  Innocent,  in  hand  and  hart :  for  trefpacing  either  againft  the  lawe  of 
God,  or  Man,  in  any  my  Studies  orExercifes,  Philofophicall,  or  Mathematical!: 

As  in  due  time,  I  hope,  will  be  more  manifeft. 

No  w  end  I, with  ArcllCIHclftriC.  Which  name,  is  not  fo  new, as  this 
Arte  is  rare.For  an  other  Arte,vnder  this, a  degree(for  fkill  and  power)  hath  bene 
indued  with  this  Englifh  name  before.  And  yet, this, may  feme  for  our  purpofc, 
fufficiendy,at  this  prefent.  This  Arte,  tgacWth  to  bryng  to  adtuall  ex¬ 
perience  fenhble,all  worthy  conclufions  by  all  the  Artes  Mathema¬ 
tical!  purpofed,  &  by  true  Naturall  Philofophie  concluded  :& both 
addeth  to  them  a  farder  fc :ope,in  the  termes  of  the  fame  Artes ,  &  al¬ 
io  by,hys  propre  Method, and  in  peculier  termes,  procedeth ,  with 
helpe  of  the  forefayd  Artes  ,  to  the  performance  of  complet  Expe- 
rieces, which  of  no  particular  Art,  are  hable  (Formally)  to  be  challen¬ 
ged  .  If  you  remember, how  we  considered  ^Architecture,  in  refpetft  of  all  com¬ 
mon  handworkes  :  fame  light  may  you  haue, therby, to  vnderftand  the  Souerain- 
ty  and  propertie  of  this  Science.  Science  I  may  call  it,rather,  then  an  Arte:  for  the 
excellency  and  Mafterlhyp  it  hath  ,  ouerfomany  ,  and  fo  mighty  Artes  and 

A.iij.  Sciences. 


1. 

2. 


lohn  Dee  his  Mathematicall  Preface. 

Sciences.  And  bycaufe  it  procedeth  by  Experiences ,and  fearcheth  forth  the  caufes 
of Conclufions,by  Experiences :  and  alfo  putteth  the  Conditions  them  felues,  in 
Experience ,it is  named  of  fom c^Scientia  Experimental^ .  The  Experimentall  Sci* 
ence.  Nicolaus  Cufanus  termeth  it  fo,  in  hy  s  Experimentes  Statikall ,  And  an  other 
Pbilefopher ,  of  this  land  Natiue  ( the  flcure  of  whofe  worthy  fame, can  neuer  dye 
nor  wither)  did  write  therof  largely,  at  the  requeft  of  Clementthe  fixt.  The  Arte 
carrieth  with  it,  a  wonderfull  Credit :  By  reafon,  it  certefieth ,  fenfibly, fully,  and 
completely  to  the  vtmoft  power  of  Nature, and  Arte.  This  Arte,certifieth  by  Ex¬ 
perience  complete  and  abfolute :  and  other  Artes,with  their  Argumentes,and  De- 
monftrations ,  perfuaderand  in  wordes,proue  very  well  their  Concluflons.  *.  But 
U*  wordes,and  Argumentes,are  no  fenfible  certifying.-  nor  the  full  and  finall  frute  of 

Sciences  praftifable.  And  though  fome  Artes,haue  in  them, Experiences, yet  they 
are  not  complete ,  and  brought  to  the  vttermofl, they  may  be  ftretched  vnto,and 
applyed  fenfibly.  As  for  exam  pie:  the  Naturall  Philofopher  difputeth  and  malceth 
goodly  fliew  of: reafon :  And  the  Aftronomer,and  the  Optical!  Mechanicien,put 
fome  thynges  in  Experience: but  neither, all, that  they  may:  nor  yet  fufficiently,  and 
to  the  vtmoft,thofe,which  they  do.  There, then,the  Archemajler  ftcppeth  in, and 
leadeth  forth  on  ,  the  Experiences ,  by  order  of  his  dodrine  Experimentall ,  to  the 
chief  and  finall  power  of  Naturall  and  Mathematicall  Artes.Oftwo  or  three  men, 
in  whom, this  Defcription  of  ArchemaBry  was  Experimentally  rifled,  I  haue  read 
and  hardrand  good  record, is  of  their  fuch  perre&ion..  So  that,this  Art,  is  no  fan- 
tafticall  Imagination:  as  fomeSophifter,  might,  Cum  fats  Infolubiltbtu^  make  a  flo- 
rifh:  and  dalfell  your  Imagination: and  dafln  your  honeffc  defire  and  Courage, from 
beleuing  thefe  thinges,fo  vnheard  of,fo  meruaylous,&  of  fuch  Importance.  Well: 
as  you  will.I  haue  forewarned  you.I  haue  done  the  part  of  a  frende.-I  haue  dischar¬ 
ged  my  Duety  toward  God:for  my  finall  Talent,  at  hys  moft  mercyfull  handes  re- 
ceiued.  T o  this  Science,doth  the  Science  Alnirangiat, great  Seruice.  Mule  nothyng 
of  this  name.  I  chaunge  not  the  name,  fo  vfed,  and  in  Print  publifhed  by  other: 
beyng  a  name,  propre  to  the  Science.  Vnderthis,commeth  <^Ars  Smtrillia ,  by 
Artefbius,  briefly  written .  But  the  chief  Science ,  of  the  Archemafter ,  ( in  this 
world)as  yetknowen  ,  is  an  other  (  as  it  were)  OPTICAL  Science :  wherof, 
the  name  finall  be  toldf  God  willyng)  when  I  finall  haue  fome,  ( more  iuftjoccafion, 
therof,  to  Dilcourfe. 

Here,I  muft  end ,  thus  abruptly  (  Gentle  frende,  and  vnfayned  louer  of  honeft 
and  neceflary  verities.)  For,they,who  haue(for  your  fake,  and  vertues  caufe)re- 
quefted  me,(an  old  forworn e  Mathematicien)  to  take  pen  in  hand :  ( through  the 
confidence  they  repofed  in  my  long  experience:  and  tryed  fincerity)  for  the  decla- 

ryng  andreportyng  fomewhat,of  thefruteand  commodity,  by  the  Artes  Ma¬ 
thematicall, to  beatteynedvnto:euenthey,  Sore  agaynft  their  willes,are 
forced,for  fundty  caufes,  to  fatiffie  the  workemans  requefl: ,  in  endyng  forthwith: 
He,  fo  feareth  this,  fo  new  an  attempt,&  fo  coftly:  And  in  matter  fo  flenderly  (he- 
therto)amongthe  common  Sorte  ofStudentes,confideredor  efiemed. 

And  where  I  was  willed,  fomewhat  to  alledge, why, in  our  vulgare  Speche,this 
part  of  the  Principall  Science  of  Geometric,  called  Euclides  Geometricall  Elementes, 
is  publifhed, to  your  handlyng  :  being  vnlatined  people,  and  not  Vniuerfitie 
Scholers :  Verily,IthinkeitnedelefTe.  ,  • 

i*  For, the  Honour,and  Eftimation  of  the  Vniuerfities,and  Graduates, 
is,  hereby,  nothing  diminiflied.  Seing,  from,  and  by  their  Nurfe  Children,  you  ' 
receaue  all  this  Benefite :  how  great  foeuer  it  be. 

Neither 


lohn  Dee  his  Mathematical!  Preface. 

N  either  are  their  Studies,  hereby,  any  whit  hindred.  No  more,  then  thxltalian  a , 

Vniuerfities, 2S  Academia  Bone  mentis,  Ferrarienfs,  Florentine,  Medielanmfis,  Patauina, 

Pagienfis,  Pentfma,  PifanayRomamfienenfis,  or  any  one  of  them,  finde  them  fellies, 
any  deale, difgraced,  or  their  St  udies  any  thing  hindred ,  by  F rater  Lucas  de  Burge > 
or  by  ‘liicelaus  T artalea,  who  in  vulgar  Italian  language, haiie  publifhed,  not  onely 
*  Euchdes  Geometric,  but  of  Archimedes  feme\vhat :  and  in  Arithmetike  and  Practical!. 
Geometric,  very  large  volumes,  all  in  their  vulgar  fpeche .  N  or  in  Germany  haue 
the  famous  Vniuerfities,  any  tiling  bene  diicontent  with  Albertns  Durerus, his  Geo¬ 
metrical!  Inftitutions  in  Dutch;  orwith  Gulklmus  Xy lander,  his  learned  tranflation 
of  the  firftfixe  bookes  oPEuclide,  outofthe  Greke  into  the  high  Dutch  .Nor  with  : 

Gmlterm  H .  Riffius ,  his  Geometricall  Volume :  very  diligently  tranflated  into  the 
high  Dutch  tounge,  and  publifhed .  Nor  yet  the  Vniuerfities  of  Spaine,  or  Portu- 
gall,  thinke  their  reputation  to  be  decayed ;  or  fuppofe  any  their  Studies  to  be  hin-  - 
dred  by  the  Excellent  P.  ?{onmus,  his  Mathematical!  workes, in  vulgare  fpeche  by 
him  put  forth .  Haue  you  not,  likewife,  in  the  French  tounge,  the  whole  Mathe¬ 
matical!  Quadriuie  ?  and  yet  neither  Paris,  Orleance, or  any  of  the  other  Vniuer¬ 
fities  ofFraunce,  at  any  time,  with  the  T ranflaters,or  P ublifhers  offended  :  or  any 
mans  Studie  thereby  hindreeb 

And  furely  ,  the  Common  and  V ulgar  Scholer  (  much  more,  the  Gramarian)  3 . 

before  his  comming  to  the  Vniuerfitie,  ihall  (  or  may)  be ,  now  (according  to  Plato 
his  Counfell)  fuiriciently  inftruded  in  Arithmetike  and  Geometrie, For-  the  better  and 
eafier  learning  ofallmaner  of  P  hilofophie,  Academically  P  erf  ateticall.  And  by  that 
meanes,  goe  more  cherefully,  more  fkilfully, and  fpedily  fonvarde,  in  his  Studies, 
there  to  be  learned.  And,fo, in  leffe  time,profite  more,then  (otherwife)  he  fliotild, 
or  could  do.  J  ••• 

Alfo  many  good  and  pregnant  Engliihe  wittes,  of  young  Gen  tlemen, and  of  4« 
other,  who  neuer  intend  to  meddle  with  the  profound  fearchand  Studie  of  Philo- 
fophie  ( in  the  Vniuerfities  t obe  learned  )  may  neuertheleffe,  now,  with  more  eafe 
and  libertie,  haue  good  occaiion ,  vertuoufly  to  occupie  the  fliarpheffe  of  their 
wittes :  where,els  (perchance  )  otherwife, they  would  in  fond  exercifes,fpend  (  or 
rather  leefe)  their  time :  neither  feruing  God :  nor  furderingthe  Weale, common 
orpriuate. 

And  great  Comfort,  with  good  hope,  may  the  Vniuerfities  haue,  by  reafon  of  5 . 
this  Engltfe  Geometrie,. and  Mathematicall  Preface, that  they  (hereafter) 

Ihall  be  the  more  regarded ,  efteemed ,  and  reforted  vnto.  For,  when  it  ihall  be 
knowen  and  reported,  that  of  the  Mathematicall  Sciences  onely, fuch  great  Commo¬ 
dities  are  enfuing  (  as  I  hauefpeciiied ) :  and  that  in  dede,  fome  of  you  vnlatined 
Studentes,  can  be  good  witneile, of  fuch  rare  fruite  by  you  enioyed  (thereby)  :  as 
either,before  this, was  not  heard  of;  or  els,n0tfb  fully  credited:  Well,may  all  men 
conie&ure,  that  farre  greater  ayde,and  better  furniture, to  winne  to  the  Perfection  „ 
ofall  Philofophie,may  in  the  Vniuerfities  be  had:  being  the  Storehoufes  &  Threa-  Vniuerfities 
lory  of  all  Sciences,  and  all  Artes,  neceflaryfor  the  beft,  and  moft  noble  State  of  »» 
Common  Wealthes.  ,  „ 

Befides  this,  how  many  a  Common  Artificer,  is  there,  in  thefe  Reaimes  of  6. 
England  and  Ireland,  that  dealeth  with  Numbers, Rule, &  CumpafTe :  Who  with 
their  owne  Skill  and  experience, already  had,  will  be  hable  (  by  thefe  p-00d  helpes 
and  informations)  to  finde  out, and  deuifc,new  workes, ftraunge  Engines  and  In- 
ftriimentes ; :  for  fundry  purpofes  in  the  Common  Wealth  ?  or  for°priuate  plea- 
hire .?  and  for  the  better  maintay  ning  of  their  owne  ehate  ?  I  will  not  ( therefore) 

A.iiij.  fight 


lolmDee  his  Mathematical!  Preface. 

fightagainft  myne  owne  fhadowe.  For,  no  man  (lam  fure)  will  open  his  mouth 
againft  this  Enterprife.No  ma  (I  fay)  who  either  hath  Charitie  toward  his  brother 
(  and  would  be  glad  of  his  furtherance  in  vertuous  knowledge)  :  or  that  hath  any 
care  &  zeale  for  the  bettering  of  the  Comon  ftate  of  this  Realme.N  either  any, that 
niake  accompt,  what  the  wifer  fort  of  men  (  Sage  and  Stayed  )  do  thinkc  of  them; 
T o  none  ( therefore  )  will  I  make  any  Apologie,  for  a  vertuous  a&e  doing :  and  for  * 
c6mending,or  fetting  forth, Profitable  Artes  to  English  men, in  the  Englifh  toung. 
„  But,  vnto  God  our  Creator ,  let  vs  all  be  thankefull :  for  tha  i,jfs  he ,of his  Good* 

»  neSjby  his  Town  ,  and  in  his  la  if e  dome ,  hath  Created  all  thynges ,  in  Number, 

S33  35  JV aight^and  Meafure\ So,  to  vs ,  of  hy  s  great  Mercy ,  he  hath  reuealed  Meanes, 
}>  whereby,  to  attcyne  the  fufficient  and  neceflary  knowledge  of  the  forefayd  hys 
”  three  principalllnfiramentes  :  Which  Meanes ,  I  haue  abundantly  proued  vnto 
33  you,to  be  the  Sciences  and  i^Artes  fJMathematicall. 

And  though  I  haue  ben  pinched  with  firaightnes  of tyme:that,no  way, I  could 
fo  pen  downe  the  matter(in  my  Mynde)  as  I  determined :  hopyng  of  conucnient 
layfure  ••  Y et,if  vertuous  zeale, and  honeft  Intent  prouoke  and  biyng you  to  the 
readyngand  examinyng  ofthis  Compendious  treatife,I  do  notdoute,  but,as  the 
veritie  therof(accordyng  to  our  purpofe  )  will  be  euident  vnto  you  :  So  the  pith 
and  force  therof ,  will  perfuade  you  :  and  the  wonderfull  if  ute  therof, highly  plea- 
fure  you.  And  that  you  may  the  eafier  perceiue,and  better  remember ,  the  prin- 

The  Ground,  cipall  pointes ,  whereof  my  Preface  treateth ,  I  will giue  you  the  Groundplatc 
jlatt  of  this  of  my  whole  difcourfe,in  a  Table  annexed:  from  the  foft  to  the  lafl,foinewhat  Me- 
pntfaceina  thodically  contriued. 

T  able.  If  Haft,  hath  caufed  my  poore  pen, any  where ,  to  Rumble  :  You  will,  (I  am 

fure)  in  part  of  recoin  pence,  (for  my  carneft  and  lincere  good  will  to  plea- 
fure  you) ,  Coniider  the  rockifh  huge  mountaines,  and  the  perilous 
vnbeaten  wayes, which  (  both  night  and  day ,  for  the  while  )  it 
hath  toyled  and  labored  through,to  bryng  you  this  good 
N ewes, and  Comfortable  profe, of  Vertues  frute. 

So, I  Commityou  vnto  Gods  Mercyfuil  diredion ,  for  the  reft :  hartety 
befechy  ng  hym,  to  profper  your  Studyes,and  honeft  Intentes: 
to  his  Glory, &  the  Commodity  of  our  Countrey.  Amen. 

Written  at  my  poore  Houfe 
At  Cfytortlake. 

Anno. i  s  7  o.  February.# . 


f.vee. 


J  «  IJ  C  Ga 

Here  liauc  you(accorcling  to  my  promiffe)  the  Groundplat  of’ 

my  MAT  HEM  ATI  CALL-  Pradface:  annexed  to  Euclide  (now  firft) 
publifhed  in  our  Englilhetounge.  An,  i  570.  Febr.  3. 


I 

f  _ 

Sciences, 
and  Artes 

Mathe- 

maticalL 


Erincipdl, 
Iffbich  are  two, 
oneljy 


Arithmetike.< 


Simple, 


tenances ;  where,  an  Fmt,  is  Indiuijtble . 


^Geometric. 


-  ddiX  t ,  Which  With  aide  of  Geometric  principal,  demon  Sirateth  fome  tArithmeticall  Con- 
clufion ,  or  Turpofe, 

Simple 3Which  dealeth  With  Magnitudes,  onely :  and  demonSlrateth  all  their  properties,  pajfi* 
ens,  and  appertenances :  whofe  Torn ,  is  Indmfible . 

.  MlXt,  Which  With  aide  of  ^Arithmetike  principal! , demon  ftrateth feme  G eometricall purpefi: as 

EVCLIDES  ELEMENTbS. 


\  " 

In  thinges  Supernatu  -  ' 

r all, at  cm  all, &  Diuine: 

By  Application, Afcem 
ding. 

The  vfe 

thereof,  is  ^ 
*  either } 

In  thinges  Mathema¬ 
tical l:  Without  farther 

Application. 

J  . 

In  thinges  If  at ur all: 
both  Subflatiall,&  Ac- 
cidentall,Fifible,  &  In- 
uifible.&c.By  Applicat 
„tion:  Defcending. 

The  like  Vfes 
and  zfifippli  -= 
cations- are, 

( though  in  a 
degree  lower ) 
in  the  Artes 
Mathema¬ 
tical  Deri- 
uatiue. 


are 


^either< 


ft  1 


K  \ 

'  i 


The  names  of 

the  Principalis: 

as,< 


Arithmetike,  f Arithmetike  ofmoftvfuall  whole  Numbers:  And  of Fra&ions  to  them  appertaining. 
Vulgar  :  lohich  Arithmetike  of  Proportions. 
conJideretlA  Arithmetike  Circular. 

|  Arithmetike  ofRadicallNubers:Simple,Compoimd,Mixt  r  And  of  their  Fractions. 
^Arithmetike  of  Cosfike  Nubers :  with  their  Fractions :  And  the  great  Arte  of  Algiebar. 


"AllLengthes. 


Geometric, 

Twlgarispbich  tea. 
cbetb  Measuring' 


Athand—r  <  All  Plaines:  As,  Land,  Borde,  GlaiTe,&c.- 
w  All  Solids :  As, Timber, Stone,V elTels,&c.“ 


from  the  ib 


ace 

ng 


fDeriuatiue 

fro  tbe  Princi* 
pads:  of  Sohichy 
Jome  bane  < 


How  farre7from  the  CMeafurer  ,  anf 
thing  is:  of  him fene,on  Land  or  Water:  called 

Apomecometrie, 

-  f 

f 

1 

Hovp  high  or  deepe,  from  the  leuell 

j  of  the  'JMeafurers  Handing ,  any  thing  is: 
j  Scene  of  hym ,  on  Land  or  Water :  called 

Hypfometrie. 

Of which 

are  o-rowen 

0 

the  Feates 
>&•  Artes  of  < 

r 

Ho\ V  broad  1  a  thing  is ,  which  is  in  the 
Meafurers  vcw  :foithe  ftuated  on  Land  or 

Water ;  called  Platometrie. 

^  J 

Mecometrie. 

<J  Embadometrie, 
Stereometric. 


Geodefie :  more  cunningly  ts 
Meafure  and  Suruey  Landes a 
Woods,  Waters.&c. 


Geographic. 


Hydrographic. 
Stratarithmetrie. 

'  Which  demonfrateth  the  matters  and  properties  of  all  Radiations  SDireBc,  'Broken,  and  TfefleBed. 

Aftronomie,—  ~  Which  demonstrated  the  Distances,  Magnitudes, and  all  IfaturaH  motions, Apparencesjand  Tajfons ,  proper  to  the  Planets  and 

fixed  Starres.for  any  time,  paft,  prefent,  and  to  come :  in  refpeBe  of  a  certaine  Horizon. or  Without  refpeBe  of  any  Horizon, 

Mufike,  —  —  Which  demonfrateth  by  reafon,and  teaeheth  by fenfe, perfectly  to  iudge  and  order  the  diuerftie  ofSoundes ,  hie  or  lew. 

Cofmographie,  ■—  Which, Wholy  and  perfectly  maketh  defeription  of  the  Heautnly,andalfo  Elementall  part  of  the  World :  and  of thefe  panes, maketh 

homologall  application,  andmutuall  collation  necefary . 


Perlpectiue,- 


fPropre  names 
^  A 


Aftrologie, 
Statike,  — 


Whic  b  reafonably  demonfrateth  the  operations  and  cjfeBes  of  the  natural!  beames  of  light, and fecrete  Inf  Hence  of  the  'Planets ,and 
fixed  Starres ,  in  cuery  Element  and  Elementall  body :  at  alt  times,  in  any  Horizon  ajftgned. 

Which  demonfrateth  the  caitfcs  ofheauincs  and  light nes  of  all  t  hinges :  and  of  the  motions  and  properties  to  heauines  and  lightnes 


«[[  Imprinted  by  John  Day . 

An.  1  s  70.  Feb.z*. 

'  : 


Allthr  Op  Ogl~ap  hie,  whic  h  defribeth  the  JJnbcr,  Meafure,  Waight,  Figure,  Situation,  and  colour  of  cuery  diners  thing  contained  in  the  perfetle  body  of 

tW  zA  Ifjandgeueth  certaine  knowledge  of  the  Figure, Symmetric,  Waight,- Char  aBerization,&  due  Locallmotion  of any  per  cell 
/-p  |  -t  •  I  of the fay  d  body  afigned:  and  of numbers  to  the faid per  cell  appertaining. 

1  rOChUlKe,  — —  -  which  demonfrateth  the  properties  of  all  Circular  motions:  Simple  and  Compound. 

Helxcofophie,  — — ■—  which  demonstrated  the  defgning  of  all  Spiral!  lines :  in  Plaine,on  Gy  Under, Cone,  Sphtre,  Conoid,  and  Spharoid :  and  their  pro- 
A  perties, 

Pneumatithmie,  —  Which  demonfrateth  by  clofe  hollow  G eometricall figures  ( Regular  and  Irregular  )  the firaunge  properties  ( in  motion  or  Stay  )  of 

the  Water, ^Ayre, Smoke, and  Fire, in  their  Continuity ,and as  they  areioyned  to  the  Element es  next  them. 

Menadrie,—  — — ~  Which  demonfrateth, how,  aboue  Ifaturcs  ZSertue,  and  power f tuple :  ri)crtue  and  force, may  be  multiplied :  and  fo  to  direBe,  to 

lift,  to  pull  to,  and  to  put  or  call  fro, any  multiplied,  or fimple  determined  Vertue,  Waight,  or  Force :  naturally,  not, ft,  dircBible,  or 
I  t  •  t  •  moueable. 

Jtlypogeioaie,  *  Which  demonfrateth , how  ,vnder  the  Spharica/l  Superficies  of  the  Earth, at  any  depth,  to  any  perpendicular  line  ajftgned  (  Whofe  di¬ 
fiance  from  the  perpendicular  of  the  entrance :  and  the  Azimuth  likew’fe,  In  refpeBe  of  the fay  d  entrance, is  knowen  )  certaine  Way, 
r  T  J  „  °  mayheprefcribedaridgone,&c. 

JTiy  Q1  agogie,— —  Which  demonfrateth  the  pajfible  leading  of  Water  by  Natures  law, and  by  artificial i  helpe,from  any  head(  being  Spring,  Slanding,or 

running  Water  )  to  any  other  place  ajftgned. 

Horometrie,  —  —  Which  demonfrateth, hove, at  all  times  appointed,  the precife,vfuall  denomination  'ftime,m(ty  be  knowen, for  any  place  ajftgned, 

Zographie,  —  ■  Which  demonfrateth  and  teaeheth, how,  the  Inter feBion  of  ail  vifuall  Pyramids,  made  by  any  plain  e  ajfignedC  the  Centcr,difiancet 

and  light  es  being  determined  )  may  be, by  lines, and  proper  colours  reprefented . 

Architecture,—  Which  is  a  Science garnifed  With  many  doBrines, and diuers  InfruBions :  by  Whofe  iudgemcnt,allWorkcs  by  other  Workmen  fini- 

fied, are  fudged. 

Nauip-ation,— —  Which  demonSlrateth,  how,  by  the  Short  efl  good  Way, by  the  apteft  direBion,and  in  the  forteSl  time-.afujficient  Shippe,  betwenea - 
^  ny  tno  places  (in  pa jjage  nauigab  le)  afigned , may  be  conduBediand  m  all formes  and  naturall  disturbances  chaunctng ,  how  to  vfe 

— ..  the  beStpojfblemeanes, to  recouer  the  place firSt  ajftgned. 

i  haumaturglke,-  which genet  h  certaine  order  to  make  firaunge  Workes,ofthefenfe  to  be  perceiuedtand  of  men  greatly  to  be  Wondred  at . 

Archemaftrie,  Which  teaeheth  to  bring  to  aBuall  experience fenfble,a!l  Worthy  conclufions  ,by  all  the  Artes  Mathematicallpurpofed :  and  by  true 

Naturall  philofephie,  concluded:  And  both  addeth  to  themafarder  Scope ,  in  the  termes  cf  the  fame  Artes:  andalfo ,  by  his  proper 
Meehnd. and  in  peculiar  termes,procedeth, with  helpe  of  tbe forfayd  Artes  ,tothe  performance  of  complete  Experiences :  Which,  of  no 
particular  Arte, are  hable( Formally  )to  be  challenged. 


f 


lohn  Dee  his  Mathematical!  Preface . 


(|Tbc.  firft  booke  of  Eu- 


elides  Elementes 


N  this  first  sooKsis  intreated of  the moft 
fimple,  eafie,  and  firft  matters  and  groundes  of  Geo¬ 
metry,  as,  namely,  of  Lyncs,  Angles,  Triangles,  Pa¬ 
rallels,  Squares,  and  ParaHelogrammc s .  Firft  of they r 
definitions, fhewyng  what  they  are.  After  that  it  tea- 
chcth  how  to  draw  Parallel  lyncs,  and  how  to  forme 
diuerfiy  figures  ofthreefides5&  foure  fides, according 
to,  the  varietie  of  their  fides,  and  Angles  :  &  copareth 
them  all  with  T riangles  ,&  alfo  together  the  one  with 
the  other .  In  it  alfo  is  taught  how  a  figure  of  any 
forme  may  be  chaunged  into  a  Figure  of  an  other 
forme.  And  for  that  it  entreateth  of  thefe  moft  com¬ 
mon  and  general!  thynges ,  thys  booke  is  more  vniuerfall  then  is  the  fecondc, 
third, or  any  other,  and  therefore  iuftly  occupieth  the  firft  place  in  order  :  as  that 
without  which,  the  other  bookes  of  JEucUde  which  follow,  and  alfo  the  workes 
of  others  which  haue  written  in  Geometry,  cannot  be  perceaued  nor  vnderftan- 
ded.  And  forafmuchasallthedemonftrations  and  proofes  of  all  the  propositi¬ 
ons  in  this  whole  booke,  depended  thefe  groundes  and  principles  following,, 
which  by  reafon  of  their  playnnes  neede  no  greate  declaration,  yet  to  remoue  all 
(be  it  neuer  fa  litle)  obfeuritie,  there  are  here  fet  certayne  fliorte  and  manifeft 
expofitions  of  them,  '  . 

^Definitions. 

i,  Afigne  or  point  is  that  fWhich  hath  no  part 

The  better  to  vnderftand  what  matter  of  thing  a  figneor  point  is,yemuft  note  that 
the  nature  and  propertieof  quantitie(wherof  Geometry  entreateth  )is  to  be  deuided, 
fo  that  whatfoeuer  may  be  deuided  into  fundry  partesys  called  q.uantitie.But  a  point, 
a!  though  it  pertayne  to  quantitie,  and  hath  his  beyng  in  quantitie,  yet  is  it  no  quanti- 
tie, for  that  it  cannot  be  deuided.  Becaufe(as  the  definition  faith)  it  hath  no  partes  in¬ 
to  vv filch  it  Should  be  deuided. So  that  a  pointe  is  the  leaf;  thing  that  by  minde  and  vn- 
derftandingcanbeimagined  and  conceyLied  :  then  which,th ere  can  be  nothing  lefte* 
as  the  point  A  ifi  the  raargent. 


The  argument 

of  the  firfl 

Book.** 


T>ef!nit'm)  tf 


A 


Afign  eorpointisof  Tithagoras  Scholersafterthismanner  defined:  Apojntism 
vmtte'tohtch  bath  pofnistj.  Nubers  are  coneeauedin  mynde  without  any  forme  &  figure,  *PV***f*e*’ 
and  therfore  wi  thout  matter  wheron  to  receaue  figure,  &  confequently  without  place  agoras, 
andpofition.  Wherfore  vnitie  beyngapartofnumber,hath  nopofition,  or  determi¬ 
nate  place. Wherby  it  is  manifeft,that  number  is  more  Ample  and  pure  then  is  magni- 
tude,and  alfo  immateriall.*  and  fo  vnity  which  is  the  beginning  of  number,  is  lefts  ma¬ 
terial!  then  a  figne  or  poynt,  which  is  the  beginnyng  of  magnitude.For  a  poynt  is  ma¬ 
terial!,  and  requirethpofition  andplace,andtherby  differed!  from  vnitie. 


2« 


A  line  is  length  “Without  breadth . 


Def'nitim  of 

*  line. 


There  pertaine  to  quantitie  three  dimenfions,  length,bredth,&  thicknes,or  depth.* 
and  by  thefe  thre  are  all  quatities  meafured  &  made  known .  There  are  alfo,  according 

B,j,  ip 


I 


S <tn  other  defi¬ 
nition  of  a  line. 

A n  other. 


The  cades  of  a 
line. 


Difference  of  a 
point fro  Smtj. 

Vnitie  is  a  fart 
of  number. 

A  poynt  is  »» 
pnrt  of  quart- 
title. 


Definition  of 
A  right  line. 


Definition  of  a 
right  line  after 
Campanus. 
Definitio  therof 

after  Archi¬ 
medes. 


Defining  thertf 
After  Plato. 


An  other  def¬ 
inition. 

Another. 


to  thefe  three  dimenfion  s,  three  kyndes  of continuall  quantities  :  a  lyne,  a  fuperficies , 
orplaine,anda  body.Thefirftkynde,nameIy,alineis  here  defined  in  thefe  wordes,  <>A 
lyneis  length  without  breadth. A  point,  forthatitis  no  quantitie  nor  hath  any  partes  into 
which  it  may  be  deuided,but  remaineth  indiuifible,hath  not, nor  can  haue  any  of  thefe 
three  dimenfions.lt  neither  hath  Iength,breadth,northickenes.But  to  a  line.which  is 
the  firft  kynde  of  quantitie.is  attributed  the  firft  dimenfion,  namely,  length,  and  onely 
thatjf’orithath  neither  breadth  nor  thicknes,but  is  eonceaued  to  be  drawne  in  length 
onely, and  by  it,it  may  be  deuided  into  partes  as  many  as  ye  lift,equall,or  vnequall.Bu  t 
as  touching  breadth  it  remaineth  indiuifible.  As  the  lyne  A  B,  which  is  onely  drawer* 
iti  length,  may  be  deuided  in  the  pointe  C  equally,  or  in  the 

point  D  vnequally,and  fo  into  as  many  partes  as  ye  lift.  There  ,  _ 

are  alfo  of  diners  other  geuen  other  definitions  of  a  lyne:  as  A  c  3 

thefe  which  follow. 

__  zA  lyne  is  the  moiiyng  of  a  poynte,zs  the  motion  or  draught  of  a  pinne  or  a  penne  to  your 
fence  maketh  a  lyne, 

Agayne,<*^/  lyne  is  a  magnitude  hailing  one  onely  [pace  or  dimenfion,  namely,  length  Wantyng 
breadth  and  thickyes. 

|  T he  endes  or  limites  of  4  lyne ^re point es. 

For  a  line  hath  his  beginning  from  a  point,and  likewife  endeth  in  a  point;  fo  that  by 
this  alfo  it  is  nianifeft,that  pointes,  for  their  fimplicitie  and  lacke  of  compofition,  are 
neither  quantitie,nor  partes  of  quantitie,but  only  the  termes  and  endes  of  quantitie. 
As  the  pointes  zAy  B,  are  onely  the  endes  of  the  line  A  B ,  and  no  partes  thereof ,  And 
herein  differeth  a  poynte  in  quantitie,  from  vnitie  in  number: 

for  rhat  although  vnitie  be  the  beginningof  nombers,  and  no  _ _ ___ _ ^ 

numberfasapointis  the  beginning  of  quantitie,and  no  quan-  A  3 

titiejyet  is  vnitie  a  part  of  number.For  number  is  nothyng  els 
but  a  colle&ion  of  vnities,and  therfore  may  be  deuided  into  them,  as  into  his  partes. 
But  a  point  is  no  part  of  quantitie,or  of  a  lyne*.  nfeither  is  a  lyne  compofed  ofpointes,as 
number  is  of  v  nities  .For  things  indiuifible  being  neuer  fo  many  added  together,  can 
neucr  make  a  thing  diuifible,as  an  inftant  in  tiine,is  neither  tyme,nor  part  of  tyme,but 
only  the  beginning  and  end  oftime,and  coupleth  &ioyneth  partes  of  tyme  together. 


4  A  right  lyne  is  that  "Which  lieth  equally  betwene  his  pointes. 

As  the  whole  line  zA B  lyeth  ftraight  and  equally  betwene  thepoyntes  AB  without 
any  going  vp  or  comming  downe  on  eyther  fide. 

Campanus  and  certain  others, define  a  right  find  thus:  A  _ 

A  right  line  is  thejhortefl  extenfion  or  draught, that  is  or  may 
b'e  from  onepoynt  to  an  other.  zArchimedes  defilieth  it  thus. 

A  right  line  is  the  fnorteif  of  all  lines, which  haue  one  and  the fitlf fame  limites  or  endes:  which  IS 
in  maner  al  one  with  the  definitio  of  Campanus.Ks  of  all  thefe  lines  A  B  C7A  D  C7A  E  Cs 
A  F  C,  which  are  all  drawen  from  the  point  A7  to  the 

poynte4?£as  Campanus  fpeaketh,  or  which  haue  the  - - & 

felf fame  limites  or  endes,as  Archimedes  fpeaketh,the  t>  _ _ 

lyne  AB  C,  beyng  a  right  line,is  the  Ihorteft.  g 

tP/Wf<?defineth  a  right  line  after  this  maner:  Aright 
line  is  that  whofe  middle  part JhadoWeth  the  extremes.  As  if  f 

you  put  any  thyrig  in  the  middle  of  a  right  lyne,you  lhall  not  fee  from  the  one  ende  to 
the  other,whxch  thynghappeneth  not  in  a  crooked  lyne.  The  Ecclipfe  of  the  Sunneffay 
Aftronomers)  then  happeneth,when  the  Sunne,the  Moone,  &our  eye  are  in  one  right 
line.For  the  Moone  then  being  in  the  midft  betwene  vs  and  the  Sunne,  caufeth  it  to  be 
darkened.Diuer  s  other  define  a  right  line  diuerfly,as  followeth, 
j tA  right  lyne  is  that  which  fiandeth  firme  betwene  his  extremes. 

Aga  ync,A  right  line  is  that  which  With  an  ether  line  of  lyke forme  cannot  make  a  figure. 

Agayne, 


of Euclides  Elementes .  FoL  2. 

A gay  ne,<*^  right  lyne  it  that  which  hath  not  one  pan  in  a  plain:  fuperficies,  and  an  other  erected, 
on  high. 

-  Aeayne,  Aright  lyne  is  that, all  Whofi partes  agree  together  With  all  his  other  partes, 
Agayne,-^  right  lyne  is  that,whofe  extremes  abiding, cannot  be  altered. 

Euclide doth  not  here  define  a  crooked  lyne, for  it  neded  not.lt  may  eafely  be  vnder- 
ftand  by  the  definition  of  a  right  lyne,  for  euery  contrary  is  well  manifefted  &  fet  forth 
by  hys  contrary. One  crooked  lyne  may  be  more  crooked  then  an  other, and  from  one 
poynt  to  an  other  may  be  drawen  infinite  crooked  lynes :  but  one  right  lyne  cannot  be 
righter  then  an  other,  and  therfore  from  one  point  to  an  other,  there  may  be  drawen 
but  one  right  lyne. As  by  the  figure  aboue  fet,you  may  fee, 

5  yfftfper fries  is  thatftobicb  bath  onely  length  and  breadth. 


r  A  fuperficies  is  the  fecond  kinde  of  quantirie,  and  to  it  are  attributed  two  dimenfi- 
ons,  namely  length,  and  breadth.  As  in  the  fuperficies  <sArBCcD, 
whofe  length  is  taken  by  thelyne^A,  or  CD,  and  breadth  bythe 
lyne  ^4C.or‘2?£Z):andbyreafonofthofetwodimenfions  a  fuper¬ 
ficies  may  be  deuided  twowayes,  namely  by  his  length,  and  by  hys 
breadth, but  not  by  thicknefie/orit  hath  none.For,that  is  attribu¬ 
ted  onely  to  a  body,which  is  the  third  kynde  of  quantitie,and  hath 
all  three  dimenfions,length,breadth,  and  thicknes,and  may  be  de- 
uided  according  to  any  of  them. 

Others  define  a  fuperficies  thus :  A fuperficies  is  the  terme  or  ende  of  a  body.  As  a  line  is  the 
ende  and  terme  of  a  fuperficies, 

6  Extremes  of a  fuperficies, are  lynes. 


As  the  endes,limites,or  borders  of  a  lyne,are  pointes,inclofing  the  line:  fo  arc  lines. 
thelimites,borders,andendesinclofingafuperficies.  As  in  the  figure  aforefayde  you 
maye  feethe  fuperficies  inclofed  with  fou  re  lynes.  The  extremes  or  limites  of  a  bodye, 
are  fuperficieiles,And  therfore  a  fuperficies  is  of  fomc  thus  defined;  A  fuperficies  is  that , 
Which  endeth  or  inclofeth  a  body :  as  is  to  be  fene  in  the  fides  of  a  die,  or  of  any  other  body , 

7  flame  fuperficies  is  thatftobich  lieth  equally  betwene  his  lines . 


As  the  fuperficies  AB  CD  lyeth  equally  and  fmoothe  betwene 
the  two  lines  AB,  and  CD:  or  betwene  the  two  lines  AC ,  and 
*B  'D :  fo  that  no  part  therof  eyther  fwelleth  vpward,or  is  depref- 
feddownward.Andthisdefmitiomuchagreeth  with  the  defini¬ 
tion  of  a  right  line,  A  right  line  lieth  equally  betwene  his  points, 


A 


B 


and  a  plaine  fuperficies  lyeth  equally  betwene  his  lynes.  Others  define  a  plaine  fuper¬ 
ficies  after  this  maner: 

tyi  plaine  fuperficies, is  the  fliortesl  extenfion  or  draught  from  one  lyne  to  an  other  dike  as  a  right 
lyne  is  the  (horteft  extenfion  or  draught  from  one  point  to  an  other, 

Euclide  alfo  leaueth  out  here  to  fpeake  of  a  crooked  and  hollow  fuperfic  ies,becaufe  it 
may  eafely  be  vnderftand  by  the  diffinition  of  a  plaine  fuperficies, being  hys  contrary. 
And  euen  as  from  one  point  to  an  other  may  be  drawen  infinite  crooked  lines,  &  but 
one  right  line, which  is  the  (hortefi: :  fo  from  one  lyne  to  an  other  may  be  drawen  infi¬ 
nite  croked  fuperficielfes,&  but  one  plain  fuperficies,  which  is  the  (horteft.Here  mu  ft 
you  confider  when  there  is  in  Geometry  mention  made  of  pointes,lmes,circles,trian- 
gles,or  ofany  other  figure$,ye  may  notconceyue  of them  as  they  be  in  matter,  as  in 
woode,  inmettall,  in  paper,  or  in  any  fuchlyke, for  fo  is  there  no  lyne, but  hath  feme 
breadth, and  maybe  deuidedmor  points,but  that  fhal  haue  fomepartes,  and  may  alfo 
be  deuided,and  fo  of  others,But  you  muft  conceiue  them  in  mynde,plucking  them  by 
imagination  from  all  mattcr,fo  {hall  ye  vnderftande  them  truely  and  perfectly, in  their 
owne  nature  as  they  are  defined, As  alynetobelong,andnotbroade:andapoynte  to 

B.  i j,  b« 


Another 

An  ether, 
jin  ether. 
Vrhj  Euclid* 
here  defneth 
net  a  crooked 
lyne. 


Definition  of  A 
fuperficies. 

A  fuperfic  set 
may  he  deusded 
two  mujes. 


An  other  dofinE 
tiers  of  a  fsper- 
ficies. 

T he  extremes  of 
*  fuperficies. 


Another  de fin's « 
tionofn  fuper - 
ficies. 

Definition  of  a. 
plume  fuperfic 
cies , 


Another  definE 
t ion  of  n  pluynt 
fuperficies. 


NOTE , 

1 


Amthtr  defini¬ 
tion  of  a  flay  no 
fiber  fetes , 

An  other  defi¬ 
nition. 

Another  defi¬ 
nition. 

An  other  defi¬ 
nition. 

Definition  of  a 
playne  angle. 


definition  of  a 
r edit  lined  an- 

gk* 


Three  fndes  of 
an  pies. 

O 

VVhat  a  right 
angle  What 
alfo  a  perpendi¬ 
cular  lyne  it. 


TV  hat  an  o&- 
Sttfe  angle  it. 


Si*?  The first  Tioolg 

be  fo  little, that  it  fhall  haue  no  part  at  all. 

Others  othcrwyfe  define  a  playne  fuperfic ics'.cyf  plains  Jkperficies  is  that,  which  is  firmly 
fit  betxvene  his  extremes, as  before  wa  s  fayd  of  a  right  lyne. 

Agayne.e^f  plaine Jkperficies  is  that  junto  all  Whofi  partes  a  right  line  may  Well  be  applied. 

Again,  A  plaine  faperficies  is  that , Which  is  the fhorteft  ofal Jkperficies, which  haue  one  &  the  felf 
extremes:  Asa  right  line  was  the  fiiortefi;  line  that  can  be  drawen  betwene  two  pointes, 

Againe,Af  playne  Jkperficies  is  that,  whofi  middle  darkeneth  the  extremes,  as  was  alfo  fayd  of 
a  right  lyne. 

*  1 1 

8  A  plaine  angle  is  an  inclination  or  bowing  of  fib  o  lines  the  one  to  the  other 
and  the  one  touching  the  other ^and  not  beyng  direBly  ioyned together. 

-  As  the 
two  lines 
AB,ScB 
C, incline 
the  one 
to  the  o- 
ther,and 
touch  the 

one  the  other  in  the  point®,  in  which  point  by  rcafon  of  the  inclination  of  the  fayd 
lines, is  made  the  angle  A  B  C.  But  if  the  two  lines  which  touch  the  one  the  other,be 
without  allinclinationof  the  one  to  the  other,artd  be  drawne  dire&ly  the  one  to  the 
other,then  make  they  not  any  angle  at  all,as  the  lines  CD,  and  D  Ey touch  the  one  the 
other  in  the  point  D}  and  yet  as  ye  fee  they  make  no  angle. 

9  And  if  the  lines  18?  hich  containe  the  angle  be  right  lynesjhen  is  it  called  4 

rightlyned  angle . 

As  the  angle  A  B  C,in  the  former  figures,is  a  rightlined  angle,  becaufe  it  is  contai¬ 
ned  of  right  lines :  where  note,that  an  angle  is  for  the  moft  part  deferibed  by  thre  let- 
ters,of  which  the  fecond  or  middle  letter  reprefenteth  the  very  angle,  and  therfore  is 
fet  at  the  angle. 

By  the  contrary,a  crooked  lyned  angle,is  that  which  is  contained  of  crooked  lines* 
which  may  be  diuerfiy  figured.  Alfo  a  inixt  angle  is  that  which  is  caufed  of  them  both* 
namely,  of  a  right  line  and  a  crooked,  which  may  alfo  be  diuerfiy  figured,  as  in  the  fi¬ 
gures  before  fet  ye*may  fee. There  are  of  angles  thre  kindes,  a  right  angle,an  acute  an- 
gle,andan  obtufe  angle, the  definitions  of  which  now  follow. 

io  When  a  right  line J landing  Upon  a  right  line  maketh  the  angles  on  either 

fide  e quail:  then  either  of  tbofe  angles  is  a  right  angle.  And  the  right  lyne 
iphkh  flandeth  ere cle  is  called  a  perpendiculer  line  to  that  fifion  Tehicb 
it  fiandeth. 

As  vpon  the  right  line  CD,  fuppofe  there  do  fiand  an  other  line  A. 

A  A,  in  fuch  fort,thatit  maketh  the  angles  on  either  fide  therof  e- 
quall :  namely,the  angle  ABC  on  the  one  fide  equall  to  the  afigle 
AB  Don  the  other  fide :  then  is  eche  of  the  two  angles  A  B  C^and 
A  BID  a.  right  angle, and  the  line  A  B, which  fiandeth  ere&ed  vpon 
the  line  CD,  withoutinclination  to  either  part  is  a  perpendicular 
line,commonlycalledamongartificersaplumbelyne.  c  ft 

1 1  An  obtufe  angle  is  that  which  is  greater  then  a  right  angle. 

As 


ofEuclides  Elementes .  FoL^, 

As  the  angle  CBE  in  the  example  is  anobtufe  angle,  for  it  is 
greater  then  the  angle  A  BC,  which  is  a  right  angle, becaufe  it  con 
tayneth  it,and  containeth  moreouer  the  angle  ABE . 

12  An  acute  angle  is  that &hich  isle  fie  then  a  right  angle. 

As  the  angle  EB  Din  the  figure  before  put  is  an  acute  angle,for 
that  it  is  lefle  then  the  angle  A  B  T), which  is  a  right  angle,  for  the  right  angle  contai¬ 
neth  it,arid  moreouer  the  angle  ABE. 

i$  A  limite  or  termejs  the  ende  ofeuery  thing . 

For  as  much  as  ofthinges  infinite  (as  TUte  faith)  there  is  no  fcience,  therefore  muft 
magnitude  or  quantitie(wherof  Geometry  entreateth) be  finite,and  haue  borders  and 
limites  to  inclofeit/which  are  here  defined  to  be  the  endes  therof.  As  a  point  is  the  li¬ 
mite  or  terme  of  a  line,becaufe  it  is  thend  therof :  A  line  likewife  is  the  limite  &  terme 
of  a  fuperficies :  and  likewife  a  fuperficies  is  the  limite  and  terme  of  a  body,as  is  before 
declared. 


14  A  figure  is  that  which  is  contayned  finder  one  limite  or  terme, or  many . 


As  the  figure  A  is  contained  vnder  one  limit, 
which  is  the  round  line,  Alfo  the  figure  3  is  con 
tayned  vnder  three  right  lines.  And  the  figure  C 
vnder  foure,and  fo  of  others,  which  are  their  li- 
mites  ortermes. 


15  A  circle  is  a plaine  figure  ^conteyned  finder  one  line ,  "which  is  called  a  cir <* 
cumfierencefmto  fiahich  all  ly^ard%en  from  one  poynt  trithin  the  figure 
and  falling  fipon  the  circumference  therof are  equal!  the  one  to  the  other. 


As  the  figure  here  fetis  a plaine  figure,  thatis  a  figure  without  groffenes  or  thick- 
nes,and  is  alfo  contayned  vnder  one  line,namely,the  crooked  lyne 
B  CD,which  is  the  circumference  therof,  it  hath  moreouer  in  the 
middle  therof  a  point,  iiiamelyjthgpointe^,  from  which,  all  the 
lynes  drawen  to  the fiip®ISSeS,are  equal:  as  the  lines  AB,AC>  A 
i>7  and  other  how  many  foeuer. 

Ofall  figures  a  circle  is  the  moft  perfect,  andtherfore  is  it  here 
firft  defined, 

16  And  that  point  is  called  the  centre  of  the  circle ,  as  is  the  point  A,  which  is 
Jet  in  the  middes  of the  former  circle . 

For  the  more  eafy  declaration,that  all  the  lines  drawen  from  the  centre  of  the  circle 
to  the  circumference,are  equall, ye  muft  note, that  although  a  line 
be  not  made  ofpointes:  yet  a  point,by  his  motion  or  draught, de- 
feribeth  a  line,  Likewife  a  line  drawen,or  moued,  deferibeth  a  fu¬ 
perficies:  alfo  a  fuperficies  being  moued  maketh  a  folide  or  bodie. 

Now  the  imagine  the  line  A  3,  (the  point  A  being  fixed)to  be  mo* 
ued  about  in  a  plaine  fuperficies,drawing  the  point  B  continually 
about  the  point  till  it  returne  to  the  place  where  it  began  firft 

tomoue:  fo  (hall  the  point  2?,by  this  motion,  deferibethe  circum¬ 
ference  of  the  circlejand  the  point  tA  being  fixed, is  the  centre  of  the  circle.  Which  in 

BJii.  ali 


What  an  aiuft 
angle  is. 


T he  limite  of 
anj  thing. 

No  fcience  ef 
t hinges  infinite 


Definition  of  a 
figure. 


Definition  of  it. 
circle. 


A  circle  the 
■moft  perfeti  ef 
all  figures. 

The  centre  ef  & 
circle. 


Definition  of* 
diameter. 


Definition  of 
fsmicircl e. 


Definition  of  it 
fction  of  a  cir¬ 
cle. 


Definition  of 
recltlmed fi¬ 
gures. 

Definition  of 
three fidcd fi¬ 
gures. 


5^  The firU  TSoolfe 

all  the  time  of  the  motion  oftheline,hadlikediftance:  from  the  circumference, name¬ 
ly  ,the  length  of  the  line  A  B.  And  for  that  al  the  lines  drawn  from  the  centre  tothe  cir** 
cumlerence  are  defcribed  of  that  line,they  are  alfo  equal  vntoit,  &  betwene  thefelues. 

17  A  diameter  of  a  circle  fis  a  right  line^hich  dralben  hy  the  centre  thereof, 
and  ending  at  the  circumference  on  either  fide,  deuideth  the  circle  into 
tTVo  equall partes , 

As  the  line  B  A  Cm  this  circle  prefent  is  the  diameter,becaufe 
it  paffeth  from  the  point  3,  of  the  one  fide  of  the  circumferece, 
to  the  point  C,on  the  other  fide  of  the  circumference,  &  paffeth 
alfo  by  the  pointed  being  the  centre  of  the  circle.  Andmoreo-  3 
uer  it  deuideth  the  circle  into  two  equall  partes.-  the  one,name~ 

1 y  B  ©  C,being  on  the  one  fide  of  the  line,  &  the  other  namely, 

BE  C,  on  the  other  fide,which  thing  did  Thales  Miletius  (which 
brought  Geometry  out  of  Egiptinto  GreceJ  firft  obferue  and 
proue,  For  if  a  line  drawen  by  thecentre,do  not  deuide  a  circle  into  two  equal  partes: 
all  the  lines  drawen  from  the  centre  to  the  circumference  Ihould  not  be  equall* 


T> 


IS  A femicircle fis  a  figure  Tvhich  is  contaynedvnder  the  diameter y  andvni 
der  that  part  of  the  circumference  which  is  cut  of by  the  diametret 


As  in  the  circle  ABCTt  the  figure  B  AC  is  a  fcmicircle,becaufe 
it  is  contained  of  the  right  line  B  CC,  which  is  the  diametre,and 
of  the  crooked  line  B  A  C,  being  that  part  of  the  circumference, 
which  is  cut  of  by  the  diametret  G  C.  So  likewife  the  other  part 
of  the  circle,namely  B  D  C,  is  a  fcmicircle  as  the  other  was. 


19  A  fe  Elton  or  portion  of  a  circle ,  is  a  figure  Tvhiche  is  contayned  Vnder  a 
rightly  tie,  and  a  parte  of  the  circumference  0  greater  or  leffe  then  the 
fimicircle. 


As  the  figure  B  C,  in  the  example,  is  a  fedion  of  a  circle, & 
is  greater  then  halfe  a  circlc,and  the  figure  A  2)  C,is  alfo  a  fedi- 
on  of  a  circle,and  is  leffe  then  a  femicircle.  A  fedion, portion,  or 
part  ofa  circle  is  all  one,  and  fignifieth  fuch  a  part  which  is  ei¬ 
ther  more  or  leffe,  thenafemicircle:  fo  that  a  femicircle  is  not  ^ 
here  called  a  fedion  or  portion  of  a  circle,  A  right  lyne  drawen 
from  one  fide  of  the  circumference  of  a  circle  to  the  other,  not 
pfdlyngby  the  centre,  deuideth  the  circle  into  two  vnequall 
partes,  which  are  two  fedions,of  which  that  which  contayneththe  centre  is  called  the 
greater  fedion,and  the  other  is  called  the  leffe  fedion.  Asinthecxamplc,thepart  of 
the  circle^/  B  C,which  containeth  in  it  the  centre  E,  is  the  greater  fedion,beinggrea 
ter  then  the  halfe  circle:  the  other  part, namely  <tA  D  C,  which  hath  not  the  centre  iia 
it,is  the  leffe  fedion  of  the  circle,bcing  leffe  then  a  femicircle. 


20  (bright  lined  figures  are  fuch  tbhich  are  contaynedvnder  right  lynes. 


As  are  fuch  as  followeth,of  which  fome  are  contayned  vndcr  three  right  lines,  fom$ 
vnder  foure,fome  vnder  fiue,and  fome  vnder  mo, 

2 1  Tbre  ftded  figures ,or  figures  of  threfy  desire  fuch  Tvbich  are  contay* 

ned  Vnder  three  right  lines ,  As 


ofSuclides  Elementes . 

As  the  figure  in  the  example  A  B  C,  is  a  figure  of  three  fides, 
becaufeitis  cotamed  vnder  thre  right  lines,  namely,vnder  the 
lines  lABfB  C,CzA. 

A  figure  of  three  fides,  or  a  triangle,  is  the  firft  figure  in 
order  of  all  right  lined  figures ,  and  therfore  of  all  others  it  is 
firft  defined.  For  ynderleffe  then  three  lines,  can  no  figure  be 
comprehended. 


22»  Foure fided figures  or  figures  offoure fides  are finch ,  lehich  are  contained 
ynder foure  right  lines. 

As  the  figure  here  fet,is  a  figure  offoure  fides,for  that  it  is  c5~  .  ^ 

prehqnded  vnder  foure  right  lines,namely,A  B,B  D,D  C,C  A.  — — * - — - 

Triangles,and  foure  fided  figures  ferue  commonly  to  manyv- 

fes  in  demonftrations  of  Geometry .  Wherfore  the  nature  and  L- - - - — - 

properties  of  them, are  much  to  be  obferued,  the  vfe  of  other  ft-  *“ 
gures  is  more  obfeure. 


Foly. 

A 


23*  Many  fided  figures  are finch  "Which  haue  mo fides  then foure. 

Right  lined  figures  hauing  mo  fides  then  fo\ver,by  continual  adding  of  fides  may  be 


infinite.  Wherfore  to  define  them  all  feueralIy,accordingto  the  number  of  their  fides, 
fliould  be  very  tedious,or  rather  impoifible. Therfore  hath  Euchde  comprehended  the 
vnder  one  name,and  vnder  one  diffmition :  calling  them  many  fided  figures,  as  many 
as  hauc  mo  fides  then  foure  :  as  if  they  hauefiuc  fides,fixe,  feuen,  or  mo.  Here  noteye9 
thateuery  rightlined  figure  hath  as  many  angles,asit  hath  fides,&takcth  his  denomi¬ 
nation  afwell  of  the  number  of  his  angles, as  of  the  number  of  his  fides-  As  a  figure  co- 
rained  vnder  three  right  lines,of  the  number  of  his  three  fides, is  called  a  thre  fided  fi¬ 
gure  :  euen  fo  of  the  number  of  his  three  angles, it  is  called  a  triangle.  Likewife  a  figure 
contained  Vnder  foure  right  lines,by  reafon  of  the  number  of  his  fides, is  called  a  foure 
fided  figure  :  and  by  reafon  of  the  number  of  his  angles,  ki$  called  aquadrangled  fi* 
gure,andfo  ofothers. 


24.  Of  three fided  figures  or  triangles }  an  equilatre  triangle  is  that}  -tohich 
hath  three  equall fides. 

• »  » .  :  ,  •  . 

Triangles  haue  their  differencespartlyof  their  fides,  and 
partlyoftheirangles  .  As  touching  the  differences  of  their 
fides, there  are  three  kindes.For  either  all  thre  fides  of  the  tri¬ 
angle  aj-e  eqqalbor  two  onely.are  equall,  &  the  third  vnequal : 

©r  eis  all  three  are  vUea  uali  the  one  to  the  other.The  firft  kind 
of  triangles,namely,that  whichhath  three  equall  fidesfis  molt 

fitnple,andeafieft  to  be  knowemandis  here  firft  defined,  and 

•  •  •  • 

Ban;*. 


Definition  of 
foure fided 
figure! . 


Definition  if 
m.tnj  fided 
figures* 


Definition  of 
a  ntefUtUtSF 
Hrianglt, 


Definition  of 
«m»  t fio fee  let. 


Definition  of 
■i*  ScnUnttm. 


An  Qrtbigoni- 
vin  t<ri,tnfie. 


An  Oxlgont* 
Ifnttrfangie* 


ThefirH'Booke 


•  is  called  an  equilater  triangle, as  the  triangle  *A in  the  example, all  whofe  fides  are  of 
one  length.  .  , 

■"  )  •  ■  .  . 'W-'V-';,'  "  f* 

25 .  Ifofceksjs  a  triangle ftohich  hath  onelj  two  fides  equaU. 

The  fecond  kinde  of  triangles  4 
hath  two  fides  of  one  length, but 
the  third  fide  is  either  longer  or 
ihorter  the  the  other  two,  as  are 
the  triangles  here  figured ,B,C,D 
In  the  triangle  F,the  two  fides 
A  E  and  E  F are  equal  the  one  to 
the  other,and  the  fide  A  F, is  16- 
ger  then  any  of  them  both:Likewifein  the  triangle  Cthe  two  fides  G 
quail, and  the  fide  G  K is  greater.  Alfo  in  the  triangle  D,the  fides  L  MnndAlN3aie «. 
quail, and  the  fide  L  Nis  Ihorter. 

26.  S calenum  is  a  triangle fWbofie  three fides  are  alh>nequaU. 

As  are  the  triangles  E,F,  in  which  there  is  no 
one  fide  equall  to  any  of  theother.For  in  the  tri¬ 
angle  F,the  fide  AC  is  greater  then  the  fide  2?  C, 
and  the  fide  B  Cis  greater  the  thefide  AB.  Like- 
wife  in  the  triangle  F,the  fide  D  H,is  greater  the 
the  fide  B  (?,and  the  fide  D  G}is  greater  then  the 
fide  G  H, 


warn* 


An  Amtligoni- 
am  triangle. 


2  7.  Againe  of  triangles ,an  Orthigonium  or  a  rightangled  triangle s  l 
angle  ’Which  hath  a  right  angle. 

As  there  are  three  kindes  of  triangles,  by  reafon  of  the  diuerfitie 
of  the  fides  Jo  are  there  alfo  three  kindes  of  triangles,by  reafon  of 
•the  varietie  of  the  angles.  For  euery  triangle  either  containeth  one 
right  angle,  &  two  acute  angles :  or  one  obtufe  angle,&  two  acutei 
orthree  acute  angles  :  foritisimpoffiblethatone  triangle  fhould 
containe  two  obtufe  angles,or  two  right  angles,  or  one  obtufe  an. 
gle,and  the  other  a  right  angle.  All  which  kindes  arc  here  defined. 

Fir  ft  a  rightangled  triangle  whichehath  in  it  a  right  angle .  As  the 
triangle  B  CD,of  which  the  angle  B  CDfis  a  right  angle. 

28.  An  ambligonium  or  an  obtufe  angled  triangle $  is  a  triangle  "Which  hath 
an  obtufe  angle. 

A  sis  the  triangle  B,  whole  angle  AC 
I>,  is  an  obtufe  angle,  and  is  alfo  aScale- 
non.hauing  his  three  fides  vnequall :  the 
triangle  £,  is  likewife  an  Ambligonion* 
whofe angle  EG H, is  an  obtufe  angle, 

&  is  an  Ifofceles,  hailing  two  ofhis  fides 
equalbnamelyF  (JandG  H. 

2i).  An  oxigonium  or  an  acuteangled  triangle 3  is  a  triangle  "Which  hath  aU 

his  three  angles  acute . 

<6  _  » 

/te 


mentes. 


Foil. 


juarc* 


As  the  triangles  A, 71, CM  the  example, al 
whofe  angles  are  acute.-of  which  A  is  an  e- 
quilater  triangle,  2?,  an  Ifofccles,  and  C  a 
Scalenon.An  cquilater  triangle  is  moftfim 
pie,  and  hath  one  vniforme  conftrudrion, 
and  therfore  all  the  angles  ofit  arecquall, 
and  neuer  hath  in  it  either  a  right  angle, or 
an  obtufe:  but  the  angles  of  an  Ifofceles  or  a  Scalenon ,  may  di- 
uerlly  vary.  It  is  alfotobe  noted  that  in  comparifon  of  any  two 
fides of a  triangle,  the  third  is  called  a  bafe.  As  of  the  triangle 
ABCin  refpeft  of  the  two  lines  A B and  A C,the  line B  C,is  the 
bafe :  and  in  refpedt  of  the  two  fides  A  C  and  C  B,  the  line  A  B,is 
the  bafe,  and  likewyfe  in  refpedtof  the  two  fides  CB  &  B  A,  tfie 
line  A  C,is  the  bafe. 


5  o  Of  foure  fyded figures,  a  quadrate  orfquare  is  that,  Tthofe  fydes  are  e*  °fs 

quail 3and  bis  angles  right* 

As  triangles  haue  their  difference  and  varietie  by  reafon  of  their  . 
fides  and  angles:  fo  likewife  do  figures  of  foure  fides, take  their  varie¬ 
tie  and  difference  partly  by  reafon  of  their  fides,  &  partly  by  reafon 
of  their  angles,as  appeareth  by  their  definitions.Thc  four  fided  figure 
ABCD  is  afquare  ora  quadrate,  becaufeitis  a  right  angled  figure, 
al  hys  anglesare  rightangles,and  alfo  all  his  four  fides  are  cquall. 

'  •< 

31  A  figure  on  the  otic  fiyde  longer ^or  fiquarelike,  or  as fome  call  it ,4  long  nfo'Aon  of* 
fquare, is  that  ybic h  hath  right  angles, but  hath  not  equall  fydes # 

This  figure  agreeth  with  a  fquarc  touching  his  angles,  in 
that  either  of  them  hath  right  angles,  and  differeth  from  it 
onely  by  reafon  of  his  fides,in  that  the  fides  thcrofbe  not  e- 
quall,as  are  the  fides  of  a  fquare.  As  in  the  example,the  an¬ 
gles  of  the  figure  ABCD ,  are  right  angles, but  the  two  fides 
thereof  A  B ,  and  CD,  are  longer  then  the  other  two  fides 
D. 

3  2  Rhombus  (or  a  diamonde)  is  a  figure  hauing  fioure  equall fydes Jbut  it  is  De^itiott  ^ 

not  right  angled.  Diamond  figure 


t 


This  figure  agreeth  with  a  fquare,  as  touching  the  equallitie  of 
lines,  but  differeth  from  itin  that  it  hath  not  right  angles,  as  hath 
the  fquare  .As  of  this  figure,the  foure  lines  AB,75C,  CD ,  D  A}  be  e- 
quall.but  the  angles  therofare  not  right  angles.  For  the  two  angles 
ABC  and  A  D  C,  are  obtufe  angles,greater  then  right  angles,  &  the 
other  two  angles  B  A  D  and  BCD ,  are  two  acute  angles  leffethen 
two  right  angles.  And  thefie  foure  angles  are  yet  equall  to  foure  right 
angles:  for,asmuchas  the  acute  angle  wanteth  of  a  right  angle.,  fo 
much  the  obtufe  angle  excedeth  a  right  angle. 


a 


ffifiombaides 


'■I  V' 


Definition  of# 
eltamondUke  fi¬ 
gure. 


'Trapezia  or 
tub  la. 


Definition  of 
Turullell/na, 


W fiat  Petici- 
ens-are. 


Thefirsl^oo!^ 

3  |  hombaides(or  a  diamond  like) is  a  figure  ftohofe  oppofite /ides  ate  e* 

quail  find  lehofe  oppofite  angles  are  alfo  equally  but  it  bath  neither  e* 
quail fides  fior  right  angles. 

As  in  the  figure  A B  CT), all  the  foure  fides  are  not 
equall,butthe  two  fides  AB  and  CD} being  oppofite 
the  one  to  the  other,aIfo  the  other  two  fides  A  Cand 
B  Thbeingalfo  oppofite,  are  equallthe  oneto  the  o- 
ther.Likewife  the  angles  are  not  right  angles,but  the 
angles  CAB ,  and  CD  B,  are  obtufe  angles,  and  op¬ 
pofite  and  equall  the  one  to  the  other.  Likewyfe  the 
angles  A  B  ‘Z>,and  A C  D, are  acute  angles,and  oppo- 
fite,and  alfo  equall  the  one  to  the  other. 

34  dll  other  figures  of fours  fides  bejides  thejeyare  called  trapezia  fir  tables „ 

Such  are  all  figures,in  which  is  obferued  no  equallitie  of  fides 

nor  angles:  as  the  figures  A  andi?,in  themarget,which  haue  nei-  f  - - f 

ther  equall  fides, nor  equal  angles, but  are  described  at  all  aduen-  \  4  / 

ture  without  obferuation  of  order,  and  therefore  they  are  called  / 

irregular  figures. 

1 5  (parallel  or  equidifi ant  right  lines  are  fuchfitohicb 
being  in  one  and thefelfe fame ftipcr fries,  and pro* 
duced  infinitely  on  both  fides 3  doneuer  in  any  part 
i  concurre. 

As  are  the  lines  A 2?,and  C D, in  the  example. 


5^  Teti cions  or  requejles . 

1  From  any  point  to  any  point, to  dralr  a  right  line . 

After  the  definitions,  which  are  the  firft  kind  of  princip!es,now  follow  petitions, 
which  are  the  fecond  kynd  of  principles:  which  are  certain  general  fentences,fo  plain, 
&  fo  perfpicuous,  that  they  are  perceiued  to  be  true  as  foone  as  they  are  vttered,&  no 
man  that  hath  but  common  fence,can,nor  will  deny  them.  Of  which,  the  firft  is  that, 
which  is  here  fet.  As  from  the  point  ex/, to  the  point  who  wil  de- 
ny,but  eafily  graunt  that  a  right  line  may  be  drawn^For  two  points  -  4 

howfoeuer  they  be  fet,are  imagined  to  be  in  one  and  thefelfe  fame  A  B 

plaine  fuperficies,wherfore  from  the  one  to  the  other  there  is  fome 
fhorteft  draught, whiche  is  a  right  line.Likewife  any  two  right  lines  howfoeuer  they  be 
fet,  are  imagined  to  beinonefuperficies,  and  therefore  from  any  one  line  to  any  one 
line,may  be  drawen  a  fuperficie  s . 

.  :  4 

2  To  produce  a  right  line finite,  fraight  forth  continually , 

M  •  - 

As  to  draw  in  length  continually  the  right  line  AB ,  who  will 
not  graunt.?  For  there  is  no  magnitude  fo  great,  but  that  there  a.  B  c 
maybe  a  greter,nor  any  fo  litkgbut  that  there  may  be  a  kfle.And 

aline 


of  Euclides  Elemdntes .  FoL  6* 

a  line  is  a  draught  from  one  point  to  an  other,  therfore  from  the  point  B,  which  is  the 
ende  of  the  line  zA  2?,may  be  drawn  a  line  to  fome  other  point, as  to  the  point  C,  and 
from  that  to  an  pther,ana  fo  infinitely. 

5  Vpon  any  centre  and  at  any  diJlance}to  defcribe  a  circle . 

A  playne  fuperficies  may  in  compafle  be  extended  infi¬ 
nitely  :  as  from  any  pointe  to  any  pointe  may  be  drawen  a 
right  line, by  reafon  wherof  it  cdmmeth  to  paflfe  that  a  cir¬ 
cle  may  be  defcribed  vpon  any  centre  and  at  any  fpace  or 
diftance  .As  vpon  the  centre  A}and  vpon  the  fpace A  B,ye 
may  defcribe  the  circle  BC,8c  vpon  the  fame  centre,vpon 
the  diftance  A  (Z),ye  may  defcribe  the  circle  D  £,or  vppon 
the  fame  centre  ^according  to  the  diftaunce  A  F,  ye  may 
defcribe  the  circle  F  G,  and  fo  infinitely  extendyng  your 
(pace. 

4  jill  right  angles  are  e quail  the  one  to  the  other. 


v 


E 


B 


■a 


This  pcticion  is  moft  plaine,  and  oftreth  it  felfe  euen  to  the 
fence.  For  as  much  as  a  right  angle  is  caufed  of  one  right  lyne 
falling  perpendicularly  vppon  an  other,and  no  one  line  can  fall 
more  perpendicularly  vpo  a  line  then  an  other:  therfore  no  one 
right  angle  can  be  greater  the  an  othenneither  do  the  length  or 
Ihortenes  of  the  lines  alter  the  grcatnes  of  the  angleJFor  in  the 
example,  the  right  angle  A  B  C,  though  it  be  made  of  much  lon¬ 
ger  lines  then  the  right  angle  D  E  F,whofe  lines  are  much  (hotter,  yetis  that  angle  no 
greater  then  the  other. For  if  ye  fet  the  point  E  iuft  vpon  the  point  B,  then  (hal  the  line 
E  D,euenly  and  iuftly  fall  vpon  the  line  A .5, and  the  line  E  A, (hall  alfo  fall  equally  vpon 
the  line  B  C, and  fo  dial  the  angle  2)  E  F,be  equall  to  the  angle  A  B  £7, for  that  the  lines 
which  caufe  them,are  oflike  inclination. 

It  may  euidently  alfo  be  fene  at  the  centre  of  a  circle.  For  if 
ye  draw  in  a  circle  two  diameters,  the  one  cutting  the  other  in 
the  centre  by  right  angles,  ye  (hall  deuide  the  circle  into  fowre 
equall  partes,of  svhich  eche  contayneth  one  right  angle,  fo  are 
all  the  foure  right  angles  about  the  centre  of  the  circle  equall. 


5  When  a  right  line  falling  Vpon  t'tpo  right  lines  foth  make  on  one  &  the 
Jelfefamefyde ,  the  twoinVoarde angles  lefte  then  nvo  right  angles y  then 
foal  thefe  tHoo  right  lines  heyng  produced at  length  concurreon  that  part, 
in  "frhich  are  the  tibo  angles  lefe  then  two  right  angles . 

As  if  the  right  line  A  5,fall  vpon  two  right  lines, 
namely,C  D  and  E  F, fo  that  it  make  the  two  inward 
angles  on  the  one  fide,as  the  angles  D  HI  Sc  F I H, 
lefle  then  two  right  angles  (as  in  the  example  they 
do)  the  faid  two  lines  C  D,  and  E  F,  being  drawen 
forth  in  legth  on  that  part,wheron  the  two  angles 
being  lefie  the  two  right  angles  con(ift,(hal  at  legth 
concurre  and  meete  together:  as  in  the  point  -D,as 
it  is  eafie  to  fee.  For  the  partes  of  the  lines  towardes  2)  F,are  more  enclined  the  one  to 

C.if*  the 


VFhnt  cammt* 
fentences  tore. 

’Difference  be- 
twerse  petitions 
common [en¬ 
tentes^ 


Thefirfl  Hooke 

the  other,then  the  partes  of  thelines  towardes  CPare.  Wherfore  the  more  thefeparts 
are  produced  .the  more  they  fhall  approch  neare  and  neare.till  at  length  they  fhal  mete 
in  one  point.  Contrariwife  the  fame  lines  drawn  in  legth  on  the  other  fide,fbr  that  the 
angles  on  that  fide,namely,  the  angle  CHB, and  the  angle  El  A,  are  greater  then  two 
right  angles,  fo  much  as  the  other  two  angles  are  lefle  then  two  right  angles,  lhall  ne¬ 
wer  mete, but  the  further  they  are  drawen,the  further  they  fhalbe  diftant  the  one  from 
the  other. 

6  That  two  right  lines  include  not  a  fuperficies. 

If  the  lines  A  B  and  ./IC, being  right  lines,fhould  inclofe 
afuperficies,theymufteof  ncceffitie  bee  ioyned  together  at 
both  the  endes,andthe  fuperficies  mull  be  betwene  the.Ioyne 
them  on  the  one  fide  together  in  the  pointeA,  and  imagine 
the  point.#  to  be  drawen  to  the  point  C,  fo  fhall  the  line  ABi 
fall  on  the  line  A  C,and  couerit,andfo  be  all  one  with  it,  and  neuer  inclofe  a  fpace  or 
fuperficies. 


5-^  Common fentences . 


i  Thinges  equal!  to  one  and  the  felfe fame  thyng:  are  squall  alfo  the  one 
to  the  other* 

After  definitions  and  petitions.now  are  fet  common  fentences,which  are  the  third 
andlaft  kynd  ofprinciples.Which  are  certaine  general  propofitios,  commonly  known 
ofall'men.of  themfelues  moft  manifeft  &  cleared  therfore  are  called  alfo  dignities  not 
able  to  be  denied  ofany  .Peticions  alfo  are  very  manifeft,but  notfo  fully  as  are  the  ed¬ 
ition  fentences.and  therfore  are  required  or  defired  to  be  graunted.  Peticions  alfo  are 
more  peculiar  to  the  arte  whereof  they  are*,  as  thofe  before  put  are  proper  to  Geome¬ 
try:  but  common  fentences  are  general!  to  all  things  wherunto  they  can  be  applied, 
Agayne.  peticions  con fift  in  adions  or  doing  of  fomewhat  moft  eafy  to  be  done :  but 
common  fentences  confift  in  confideration  of  mynde,  butyetoffuch  thinges  which 
are  moil  eafy  to  be  vnderftanded,  as  is  that  before  let. 

As  if  the  line  A  be  equall  to  the  line  B,  And  if  the  line  C  3 

be  alfo  equall  to  the  line  By  then  of  neceffitie  the  lines  y  ' 

^andCjllialbeequaltheonetotheother.Soisitinall  ^  <- 

fuperficielfes,angles,&numbers,&inallotherthings  * - ; - *  1 — - — r* 

(of  one  kynde)  that  may  be  compared  together. 


And  if  ye  adde  equall  thinges  to  equall  thinges:  the  fbhole Jhalhe  equall. 


■B 


E 


V 


F 


As  if  theline  AB  be  equal  to  the  line  C  P>,&  to  the 

line  A  By be  added  the  line  B  p,& to  theline  C2?,be  - - 

added  alfo  an  other  line  E>  Pfoeing  equal  to  the  line  c 

B  E,(o  that  two  equal  lines, namely, 5  P,and  T>  P,be  * - 

added  to  two  equall  lynes  lAB,&  C Pbthen  foal  the 
w  hole  ly  ne  iAEy  be  equall  to  the  whole  lyne  CP,  and  fo  of  all  quantities  generally, 

I  And  if  from  equall  thinges 3ye  take  al&ay  equall  thinges:  the  thinges  re* 

may mng  fl? all  be  equall 


ofEuclides  Elementes  < 


Eolj. 

£  3 


c 


F 


As  if  from  the  two  lines  AB  and  CD,  being 
equal, ye  take  away  two  equalLlhies,namely,£  B 
and  F  D,  then  maye  you  conclude  by  this  com¬ 
mon  fentence,that  the  partes  remayning,name~ 
ly,^  F,and CF are  equall  the  one  to  the  other.’ 
and  fo  of  all  other  quantities . 

4  .And  if from  Vnequall  t  hinges  ye  take  away  equall  thinges :  the  thynges 

which  remayne f mil  he  Vnequall, 


E  3 


F 


As  if  the  lines  A  2?,and  CD,  be  vnequall,the  line  <iA  2?,  beyng  ^ 
longer  then  the  line  C  D,  &ifye  take  fro  them  two  equall  lines,  £~ 
as£F,andFD:thepartesremayning,whichare  thelines  a/FF  *— 
and  C  F,lhall  be  vnequall  the  one  to  the  oth  er,namely,  the  lyne 
%A F,lhall  be  greater  then  the  line  C  F, which  is  euer  true  in  all  quantities  whatfoeuer* 

5  And  if  to  Vnequall  thinges  ye  adde equall  thinges :  the  whole  fhall  he  Vn* 
equali 

Asifyehauetwo  vnequallines,namely,^Fthegreater,and  A  e  B 

C  F  the  lelfe, &ifye  adde  vnto  the  two  equall  lines,  EB  &FD,  ~~ 

thenmaye  ye  conclude  that  the  whole  lines  compofed  are  vn-  £ - — ■£* 

equall :  namely,  that  the  whole  lyne  <sA  2?,  is  greater  then  the 
whole  line  C  D,and  fo  of  all  other  quantities. 

6  Thinges  which  are  double  to  one  and  the  felfe  fame  thing:  are  equall  the 
one  to  the  other. 


A. 

.  t — - 


3 


As  if  the  line  &A  B  be  double  to  the  line  E  F,and  if  alfo  the 
lineCD,be  double  to  the  fame  line  EF:  themayyou  bythis 

common  fentence  conclude, that  the  two  lines  vA  i?,&  C  D,  £ _  F 

areequalltheoneto  theother.Andthisistrueinall  quanti-  c  ^ 

ties,and  that  not  only, when  they  are  double,  but  alfo  if  they  - - - - - 

be  triple  or  quadruple,  or  in  what  proportion  foeuer  it  be  of 

the  greater  inequallitie.  Which  is  when  the  greater  quantitie  is  compared  to  thelefle. 


7  Thinges  which  are  the  halfe  of  one  and  the felfe  fame  thing:  are  equal  the 

one  to  the  other. 


A 


F 


As  if  the  line  A  B  ,be  the  halfe  of  the  line  E  F,  and  if  the  lyne 
C  D,  be  the  halfe  alfo  of  the  fame  line  EF:  then  may  ye  con¬ 
clude  by  this  common  fentence,that  the  two  lines  c AB  and 
CD,  are  equall  the  one  to  the  other.  This  is  alfo  true  in  all 
kyndes  of  quantitie,and  that  not  onely  when  ir  is  a  halfe,  but 
alfo  if  it  be  a  third  ,a  q  uarter,  or  in  w  hat  proportion  foeuer  it 
be  of  the  lelfe  in  equallitie.  Which  is  when  the  lelfe  quantitie  is  copared  to  the  greater* 


p 


v, 

8  ThinoesWi 

C3 


jjjm  together:  We  equall  the  one  to  the  other . 


Such  thinges  are  fayd  to  agree  together,whiche  when  they  are  applied  the  one  to 
the  other;  or  fet  the  one  vpon  thfeother,the  one  excedeth  not  the  other  in  any  thyng, 

C.iij,  As 


What  proporti¬ 
on  of the grea¬ 
ter  inetpttalitj  k 


What  proporti¬ 
on  of  the  lejfe 
tnequahue  is , 


Whitt  a  L'ropo- 
Jirien  is. 


Propofitions  of 
two  fortes. 

What  ^  Pro- 
hit  tacit. 


WhaSa  Thee- 
remeis. 


TheftrSlBooke 

As  if  the  two  triangles  ABC,  and  DE  F,  were  applied 
theone  to  the  other, and  the  triangle  AB  C ,  were  fet  v- 
pon  the  triangle 2)  £  F,if  then  the  angle  A ,  do  iuftly  a- 
greewith  the  angle  F>,and  the  angle  S>with  the  angle  E, 
and  alfo  the  angle  C,  with  the  angle  F :  and  moreouer  if 
the  line  A  F,d o  iuftly  fall  vpon  the  line  T>  £,and  the  line 
A  C,vpon  the  line  D  F,  and  alfo  the  line  2  C,  vppon  the 
line  E  E,  fb  that  on  euery  part  of  thefe  two  triangles, 
there  is  iuft  agreement,  then  may  ye  conclude  that  the 
two  trianglesare  equalL 


9  Euery  Thlxple  is  greater  then  his  part. 

As  the  whole  is  equal  to  all  his  partes  taken  together,  fo  is  it  grea¬ 
ter  then  any  one  part  therof.Asif  the  line  CB  be  a  part  of  the  line  A  4 
'5, then  by  this  common  fentenceye  may  conclude  that  the  whole  ~ 
line  A  B,  is  greater  then  the  part,  namcly,the  the  line  CF,And  this 
is  gencrall  in  all  thinges. 


He  principles  thus  placed  tended, now  follow  the  propofitions,  which 
are  fentences  fetforth  to  be  proued  by  reafoningand  demonftrations, 
a  nd  thetfore  they  arc  agayne  repeated  in  the  end  of  the  dem  onftration. 
For  the  propofition  is  euer  the  conclufton,  and  that  which  ought  to  be 
proued.  ,  , ,  , 

Propofitions  are  of  two  fortes,  the  one  is  called  a  Probleme,  the  other  a  Theoreme.1 

AProbleme,is  a  propofition  which  requireth  fome  a<ftion,or  doing:  as  the  makyng 
of  fome  figure, or  to  deuide  a  figure  or  line,to  apply  figure  to  figure,to  adde  figures  to- 
gether,or  to  fubtrah  one  from  an  other ,to  deferibegto  inferib^ to  circumfcribe  one  fi¬ 
gure  within  or  without  another,  and  fuche  like .  As  of  the  firft  propofition  of  the  firft 
booke  is  a  probleme,  which  is  thus:  Vpon  aright  line geuennot  bang  infinite, to  defertbe  an  e-  <■ 
quilater  triangle, or  a  triangle  of  three  equallfides .  For  in  it,  befides  the  demonftration  and 
contemplation  of  the  mynde, is  required  fomewhat  to  be  done:  namely,  to  make  an 
equilater  triangle  vpon  a  line  geuen.  And  in  the  ende  of  euery  probleme,  after  the  de¬ 
monftration,  is  concluded  after  this  manner,  Which  is  the  thing,  Which  Veas  required  tobe 
done* 

A  Theoreme, is  a  propofition,which  requireth  the  fearching  out  and  demonftration 
of  fome  propertieor  pafiion  of  fome  figure:  Wherinis  onely  fpeculation  and  contem¬ 
plation  of  minde,  without  doing  or  working  of  any  thing.  As  the  fifth  propofition  of 
the  firft  booke,which  is  thu  s,zAn  lfofceles  or  triangle  of  typo  e  quail fides,hath  his  angles  at  the 

hafe,e quail  the  one  to  the  other,&c.isa.  Theoreme.  For  in  it  is  req  uired  only  to  be  pro¬ 
ued  and  made  plaine  by  reafon  and  demonftratio,  that  thefe  two  angles  be 
equai!,without  further  working  or  doing.  And  in  the  ende  of  euery 
Theoreme, after  the  demonftration  is  concluded  after  this  ma¬ 
ne  r.  Which  thyng  Was  required  to  be  demonstrated  or  proued . 


ofEuclides  Elementes .  Fol.  8 . 

54^  The  fir  ft  Trobleme .  Ehe fir  ft  Trnpofition » 

* Upon  a  right  line  geuen  notheynginfimtey  to  deficrihe  an  e* 
qnilater  triangle yr  a  triangle  of  three  equall  fide  s. 


qnilater  triangle  ynamely, a  triangle  of  three  equal l 
fdes.'N.o'to  the r fore  making  the  centre  the  point  A 
\and  the /pace  A  Bfefcribe(by  the  third  petition ) 
a  circle  B  C  "D'and  agayne  (by  the fame)  tnakyng 
the  centre  the  point  B  yand  the f pace  Bj{yde/cribe 
an  other  circle  JICB.  Andfby  tbefirfl  petition ) 
from  the  point  C,wherin  the  circles  cut  the  one  the 


-  ,, 

other  firaTb  one  right  line  to  the  point  A,and  an 
other  right  line  to  the  point  B.And  for  af much  e 
as  the  point  J.  is  the  centre  of  the  circle  CBt>^ 
therforefhy  the  if.  definition)  the  line  A  Cise* 
quail  to  the  line  A  B :  Agayne  forafmuch  as  the 
point  B  is  the  centre  of  the  circle  CAB ,  ther* 
fore  (by  the  fame  de  finition)  the  line  B  C  is  e* 
quail  to  the  line  B  A. And  it  is  proued ,  that  the 
line  AC  is  e  quail  to  the  line  A  B :  Tbherfore  ei * 
tber  of  the/e  lines  C  A  and  CBfis  equall  to  the 

line  flBxbut  tbingeslbhich  are  equall  to  one  and  the  fame  things  realfo  equall 
the  one  to  the  other  (by  thefirjl  common fentence)l»her fore  the  line  C  Ay  alfo 
is  equall  to  the  line  C  B ,  VV her fore  thefe  three  right  lines  C  AyA  By  and  B  C 
areequalthe  one  to  the  other.VVher fore  the  triagleA  B  C  is  e  qnilater  yV  her » 
fore  Vppon the  line  ABy  is  defcribed an  equilater  triangle  jdBC.  Wherfore 
Vppon  a  line  geuen  not  heingin finite y  there  is  defcribed  an  equilater  triangle % 
Which  is  the  thing  ftvbicb  'toas  required  to  be  done. 

A  triangle  or  any  other  re&ilined  figure  is  then  faid  to  befet  or  defcribed 
vpon  aline,when  the  line  is  one  of  the  fides  of  the  figure. 

This  firfl:  propofition  is  a  Pro£/<?;»e,becaufe  it  requireth  atte  or  doy ng, name; 
ly,todefcribe  a  triangle.  And  this  is  tobenoted>thateuery  Prop  fit  ion,  whether  it 
be  a  Probleme,oT  a  T  hear  eme, commonly  containeth  in  it  athi»vgenen  ,and  athingreqni* 
red  to  befiarchedont :  although  it  be  not  alwayes  fo.  And  rhe  thing  geuen, is  euer  fet  be¬ 
fore  ihething reejuiredAn  fome  propofitions  there  are  more  things  geuethen  one, 
and  mo  thinges  required  then  onejnfome  there  is  nothing  geuen  at  all, 

Movcoiier  euevy  ‘Problems &Theowtte,  fay ng  perfect  and  abfolute,  ought  to 
haue  all  thefe  partesanamely ,  F irh  the  Tropfimn, to  be  proued. Then  the  expfimn 

^hich 


Cmflruclien. 


Demonttratk® 


Thing  geuen. 

T  hwg  required 


Tropfititm . 
Exgoftten. 


Dacrmin.ttios). 

'Ccnfirucrion. 

Demwfiratiort. 

Cendufion. 


Cafe. 


The  thing  geuen 
■in  this  Pro- 
ileme. 

The  thing 
required. 

The  proportion. 
The  exposition. 
The  de terms  - 
nation. 

The  conjiructso 


The  i temonfira - 
turn. 


T he  particular 
cenclufion. 


The'vniuerfatl 

cenclufion. 

The  note  where 
hj  it  is  kgmvne 
So  he  a  Pro- 
ilcme. 

Mo  cafes  in  thjs 
grope  fit  sen. 


Three  k fades  of 
demottf ration. 


ThefirHTiooke 

which  is  the  explication  ofthe  thing  geuen.  After  that  followeth  the  determination 
which  is  the  declaration  ofthe  thing  required.Thenis  fee  the  conffruclion  of  fuche 
things  which  are  neccffary  ether  forthc  doingofthc  propofitio,  or  for  the  demo 
ftration*Afterwardfolloweththe^w<?»/?r4f/w,  which  is  the  reafon  and  proofe  of 
the  proportion*  And  la.lt  ofall  is  put  the  concluficn,^ hich  is  inferred  8c  proued  by 
the  demonflration,and  is  euer  the  propofition,Butull  thofe  partes  are  not  of  ne¬ 
ed  fit  ie  required  in  cuery  Problems  and  Theoreme,But  the  Propofition,  demon  flra- 
t ion, and conclufon, are  necrilary  partes ,8c  can  neuer  be  abfent:  the  other  partes  may 
fometymesbeaway. 

Further  in  diuerspropofitionsathere  happen  dtuers  cafes:  which  arenothing 
els, but  varictie  of  delineation  and  conftruftioiijor  chaunge  ofpofition,  as  when 
pointes, lines. fuperlicieffes.or  bodies  are  chaunged.  Which  thingeshappen  in 
diuers  proportions.  _  ' 

O  w  then  in  this  Probleme,ihcthinggeuen,\sthe  line  gene:  the  thing  required, to  be 

lerched  out  is,how  vpo  that  line  todefenbean  equilater  triangle.  The  Pro- 
pofitiafi  of  this  'Problems  is  fUpon  a  right  line geuen  not  b  eyng  in  finite  ft  o  defer  ibe  an  equilater  tri¬ 
angle. 

T he  expofition  [sfuppof  that  the  right  linegeuen  be  A B,  and  this  declarech  onely 
the  thing  geuen  ff  \iq  determination  i  s  fit  is  required  vpon  the  line  oA  B,  to  defer ibe  an  equilater 
mangle:  for  therby  as  you  fee, is  declared  onely  the  thingrequired.  The  conflruttisn  be- 
gi  line  th  at  thefe  wo  tds^TYjnV  therfore  making  the  cetret  he  point  A,&the fpacc  A  B,  defiribe 
(by  the  third petkion)a  circle &c, and  continueth  vntil  y  ou  come  to  thefe  wordes, 
forafinuch  as  the  point  A  e^c.For  thetheitoarc  deferrbed  circles  and  lines,  neccffary  e 
both  for  the  doyng  ofthe  proportion,  and  alfo  for  the  demonftration  therof, 
V  vhich  demonfirationhegumeth  at  thefe  wordes:  eAndforafmuche  as  the  point  A  is  the 
centre  ofthe  circle  CB  D  &c:  And  fo  proccdcth  till  you  come  to  thefe  wordes,  PDher- 
fore  vpon  the  line  A  Bis  deferibedan  equilater  triangle  A  B  C.  For  vntlll  y  OU  come  thether 
is, by  groundes  before  fee  and  conftruftions  had>proued,and  made  euident,  that 
the  triangle  made, is  equilater. And  then  in  thefe  wordes, therfore  vpon  the  line  eA% 
is  defribed  an  equilater  triangle  cA  B  C,  is  put  the  firft  conclufon.  For  there  are  common¬ 
ly  ineuery  propofmon  two  conclufions:  theoneperticulcr,the  othervniuerfal: 
and  from  the  firft  you  go  to  thelaft.  And  this  is  the  firft  and  perticuier  conclufi- 
on,for  that  it  concludeth,  that  vpon  the  lyne  AB  is  deferibedan  equilater  tria- 
gle,which  is  according  to  the  expofition*After  it,followeth  thelaft  and  vniuerfal 

conclufon, therfore  vpon  a  right  linegeuen  not  being  infinite  is  deferibedan  equilater  triangle.  For 
whether  the  linegeuen  be  greater  or  leflethen  thys  lyne,  the  fame  conftru&ios 
and  demonftrarions  proue  thefame  conclufion.  Laft  of  ail  is  added  this  claufe. 
Which  is  the  thing  which  'fyas  required  to  be  done:  wherby  as  we  haue  before  noted,  is  de¬ 
clared, that  this  proportion  is  zProblemeandnot  afWw.As  for  varictie  of  ca¬ 
fes  in  this  propofition  there  is  none, for  that  the  line  geuen, can  haue  no  diuerfi- 
ticof  pofition. 

As  you  haue  in  this  Probleme  fene  plainelyefetfoorthe  the  thing  geuen,  and  the 
thing  required,  rn O  r e o u e r  the  propofition, expofition, determination, conslrullion,  demon firation, 
and conchifiorfi  i eh  are  general!  alfo  to  many  other  both  Problemes  and T heorernes ) 

fo  may  you  by  the  example  therof  diftinft  cheai,andfearche  them  out  in  other 

Problemes  ,and  alfo  T  heorernes. 

This  alfo  is  to  be  noted,that  there  are  three  kyndcs  ofdemonftrasion.  The 
one  is  called  Demonflmio  a  priori  ,0V  compofmon.  The  other  is  called  Dmomhmw 

s  g  po ft  mcri* 


ofEuclitles  Elementes.  FoLp, 

.J 

#pofieriori,oT  refoltition.  And  the  third  is  ademonftracion  leadyng  to  animpof- 
fibilirie. 

A  demonftration  a  priori,  or  cofopofition  is, 'when  in  reafoning,  from  the  prin-  Detmnftratu* 
dples  and  firft  groundeSjWepaffedifcending  continually  ,till  after  many  reafons  c°m' 

made, we  come  at  the  length  to  conclude  that, which  we  firft  chiefly  entend.  And 
this  kinde  of  demonftration  vfeth  Euclide  in  his  bookcs  for  the  moft  parr* 


A  demonftration refolution  is,when  contrariwife  in  reafoning,  we 
paffe  from  the  laft  conclufion  madeby  theprcmiffes,andby  thepremtffes  of  the 
premiffesprontinually  afcending,til  we  come  to  the  firft  principles  and  grounds, 
which,  arc  indemonftrable,and  for  they  r  fimplictty  can  buffer  no  farther  refolu- 

tion*  '■  -:<c'  :  " 


D  emcnWratiorx 
a  pc.jlertori  }or 
reflation. 


A  dcmonftration  leadyng  to  an  impoffibilitie  is  thatargumenqwhofe  c5-  Dcmmfiratio ? 
clufion  is  impoffible:  that  is, when  itconcludeth  directly  againftany  principle, 
oragainft  any  propofition  before  proued  by  principles,  orpropofmons  before  ‘ //-7'  ■ 

proued, 

Premiffes  in  an  argument,are  proportions  goy ng  before  the  conclufion  premifes  what 
by  which  the  conclufion  is  proued,  th*j«re. 


Compofitionpaffethfrom  the  caufeto  the  effe£t,or  from  thingcsfimple  to 
thinges  more  compounded. Rcfolution  contrariwifepaffeth  from  thinges  com¬ 
pounded  to  thinges  more  fimple,or  from  the  effect  to  the  caufe. 


Gompofitionorthefirftkyndeofdemonftration,  whichpaffeth  from  the, 
principles,may  eafely  be  fene  in  this  firft  propofition  of  Euclide*  The  demon- 
ftration  wherofbeginneth  thus*  Andforafmuch  a-s  the  point  A  is  the  centre  of 
the  circle  C  B  D,therforetheline  AC, is  equal  to  the  line  AB.This  reaibn(yoa 
fec)takcth  hisbegtnnyng  ofa principle, namely ,ofthe definition  of  acirde.And 
this  is  the  firft  reafon.  Agayneforafmuchas  B  is  the  centreof  the  circle  C  A  E, 
therfore  the  line  B  C  is  equalltothelyne  B  A:which  is  thefecond  reafon.  And 
it  was  before  prouedthat  thelyne  A  C  is  equal!  to  the  line  A  B,  wherforeeither 
ofthefe  lines  C  A  Sc  C  B  is  equal  to  the lyne.A.3 .And  this  is the  third  reafS* .‘But 
things  which  are  equall  to  one  &  the  felfe  fame  thy  ng, are  alfo  equall  the  one  to 
the  other. Wherfore  the  line  CAis  equal.to  the  line  CB.  a  fid  this  'is  the  fourth 
argument.  VVherforethefe  three  lines  C  A,  A  B, and  B  Care  equall  the  one  to 
the  other  which  is  the  conclufion, and  the  thing  to  be  proued. 

You  may  alfo  in  the  fame;  firft  Propofido,eafely  take  an  exaple  of  Refokirid: 
vfing  a  contrary  order  pafify  ng  backward  fro  the  laft  conclufio  of  the  former  de- 
monftration,til  you  come  to  the  firft  principle  or  ground  wheron  it  began,-  For 
the  laft  argument  or  reafon  in  compofitioh.,is  the  firft  in  Refolution:  Zc  the  fir.fi: 
in  compofitlohjis  the  laft  lnxefolution.T  hu$ iherfore  muft  ye  precede*  The  tq 
flAgle  .A  B,C  is  contained  ofthree  equall  ri  ght  lines, namely  j  A  B,A  C,andfi  Cj 
and  therfore  it  is  an  equilater  triangle  by  the  definition  of  an  equilater  triangle: 
and  this  is  the  firft  reafon.  That  the  three  lines  be  equall,  is  thus  proued.  The 
line  s  A  G  and  C  B  are  equall  to  the  line  A  B, wherfore  they  are  equall  the  one  to 
the  other:  and  this  is  thefecond  reafon. That  thelines  AB  andB  C, are  equal  is 
thus  proued:  1  he  lines  A  B  a;id  A'C*  are  draw'en  from  the  centre  of  the  circle  A 
C  E,to  the  circumference .oTpReTame:  wherforethey  are  equall  by  the  dcfinitio 
ofa  circle:  and  this  is  the  third  reafon* Ukewife  char  thelines  A  C  and  A  B,  are 
»  -  ■  ■  D  *  i,  equall 


An  example  of 
conrpeftton  m 
the firji  propoft 
tsen, 

Tirjl  reafon. 
Second  reafn. 
Third  reafon* 
fourth  reafon. 
Conclufion. 


Example  of  re* 
folution  tn  the 
frrft  prcpepciofti 
ftrft  reafon. 


Second  reafo  n, 

T bird  reafon, 

fourth  reafon 
which  ts  the  end 
of  the  tv  hole  rent 
f  inti  on. 


•ft ore  to  deferibe 
txn  ifcjcelei 
£rj/i,<;g:c. 


Hftv  to  describe 
•«t  Sc  Men  urn. 


"  V~hefirftcBoofy 

equall,is  proued  by  the  fame  reafon.  For  the  lines  A  C  and  A  B  arc  draVn  froth 
the  centre  of  the  circle  B  CD:  wherforcthcy  areequallby  the  fame  definition 
oia  circle:  this  is  the  fourth  reafon  or  fillogifme*  And  thus  is  ended  the  -whole 
refolution :  for  that  yon  are  come  to  a  principle.which  is  indemoftrablc.  8£  can 
notberefolued. 


Ora  iemonfirationleadingtoan  impoffibiIitie,orto  anabfurditie,you  may 
hauean  example  m  the  fourth  proposition  ofthis  booke*. 


RVt  nowc  if  vpoft  the  fame  line  geuen,  namely,  tAB^yc  wil  deferibe  the  other  two 
kinds  of  triangles,  namely,an  ififceles  or  atriagleoftwo  equal  fides,&aScv2/f»c’w,or 
a  triangle  of  three  vnequall  hdes.Firfl  for  the  deferibing  of  an  Ifofctln  triangle  produce 
the  line AB  on  ether  fide,vntill  ic  concur  with  the  circumferences  of  both  the  circles 
in  the  pomtes  D  an  d  i^/and  making  the  centre  the  point  .^deferibe  a  circle  H  F  t/ ac¬ 
cording  to  the  quatity  pf  the  line  A  F. 


Likewile  making  the  centre  the  poynte 
i?,defcribe  acircle  HCD  G, according  to 
the  qua n tide  of  theline  B  D  .  Now  the 
thefe  circles  ihaii  cut  the  one  the  other 
in  two  poyntes ,  which  let  be  Hy and  6“: 

And  let  the  endes  of  the  line  geuen  be 
ioyned  with  one  of  the  fayd  factions  by 
two  right  linesjwhich  let  be  *X G  and  B 
G.  And  forafmuc'heas  thefe  two  lines 
AS  and  A  D  are  drawen  fro  the  centre 
of  the  circle  CDE  vntothe  circumfe- 
rece  therof,tberfore  ar  they  eq  ual .  Like 
wife. the  lines  B  A  and  B  A,  for  that  they 
are  drawen  from  thecentre  of  thccir- 
cle  E  <lA  CF  to  the  circumference  ther- , 
of, are  equal, And  forafmuch  as  ether  of 
the  lines  aA  Dznd  B  Fis  cquall  to  the 
line  ^i?,therfore  they  are  equal  theohe 
to  the  otheivWherfore  putting  the  line  A  B  comoto  the  both, the  w  hole  line  B  2)  fhal- 
b’e  equal!  to  the  whole  line  *A  F.V>i\tB  Dh  equal  to  B  G,for  they  are  both  drawen  fro 
the  cetre  of  the  circle  HD  G  to  the  circumferece  therof.Andlikewife  by  the  fame  rea- 
fen  the  line  AFis  equal  to  the  line  tA'  G.  Wherfore  by  the  como  fcntece  the  lines  A  G 
and  B  Cj  are  equal  the  one  to  the  other, and  either  of  them  is  greater  then  the  line  A  B, 
for  that  either  of  the  two  lines  B  Z>  and  Fis  greater  then  the  line  AB,  Wherfore  v?, 

pon  the  line  geuen  is  defcri’bedan  ijhfteks  or  triangle  of  two  equall  (ides. 

Ye  may  alfo  deferibe  vpo  n  the  felfe  fame  line  a  Scaleuon ,  or  triangle  of  three  vne- 
quali  fides,if  by  two  right  lines,  ye  i:oyne  both  the  endes  of  the  line  geuen  to  fome  one 
point  rhatis  in  the  circumference  of  one  of  the  two  greater  circles  ifo  that  that  poynt 
benot  in  one  of  the  two  fections,and  that  the  line  2)  F  do  not  concur  with  it,  when  it 
is  on  either  fide  produced  contlnuallye  and  dire&lyc.For  let  the  poynte  if.be  taken  in 
the  circumfcreace  of  the  circle  HD  G}  and  let  it. not  be  in  any  of  the  lections ,  neyther 
let  the  line  I?  A"  concur  with  it, when  it  is  produced  continually  and  dire&ly  vntothe 
circumfe/encetherof.  And  draw  thefe  lines  A  if  and  i?  if,  and  the  line  zA  K  jhal  cut  the 
circumference  of  the  cixdzHFG  .  Let  it  cut  it  in  the  poynte  L  :  mow  then  by  the 
common  fentence  the  line  B  K fhalbe  equal  to  the  line  aA  L,for(by  the  definition  ofa 
cirdekhelinei?  if  is  equall  to  theline  B  6',and  the  line  A  L  is  equall  to  the  line  A  d 
which  is  equal  to  the  linei?  G.Wherfore  the  line  <lA  if  is  greater  then  the  line  B  K  an  3 
by  the  lame  reafon  maye  it  be  proned  that  the  line  B  if is  greater  then  the  line  a AB. 

Wherfore 


FoLio. 

Wherfore  the  triangle  AB  K  cpnfifteth  of  three  vnequal  fides.  And  fo  haue  ye  vp6n  the 
line  geuen,defcribed  all  the  kiudes  of  triangles.  •  • 

This  fotobe  noted,  that  if  a  man  will  mechanically  and  re,.  ^ 

defy, not  regarding  demonftratson  vpon a  line  geuen  defcf'xbe 
a  triangle  of  three  eqaall  fides,  he  needethhot  to  deferibe  the 
whole  forelay d  circle, but  onely  alirtle  part  of  eche:  namely, 
wherethey  cut  the  one  the  other, and  fo  from  the  point  of  the 
fedion  to  draw  the  lines  to  theendesoftheiinp  g.euen.  as  in 
this  figure  here  pur. 

And  likewife,if  vpon  the  fa  id  line  he  will  d'efcribea  trian¬ 
gle  of  two  equal  l  fydes,  let  him  extende  thecompaffe  accor¬ 
ding  to  the  qnantitie  that  he  will  haue  the  fyde  to  be,  whether 
longer  then  the  line  geuen  orfhortcr*.  andlo.dtaw  onely  a  ii- 
tie  part  ofcchecirclc,where  they  cut  the  one  the  other,  be  fro 
the  point  of  the  fedion  draw  the  lines  to  the  eude  of  the  line 
geuen.  Asia  the  figures  hereput.  Note  that  in  this  the  two 
fydes  muftbc  fuch3thatbeyngioynedtogether3  they  be  lon¬ 
ger  then  the  line  geuen* 

And  fo  alfo  if  vpon  the  fay  d  right  line  he  will  deferibe  a 
triangle  of  three  vnequal  fydes  ,let  him  extend  the  compare. 

Firft,  according  to  thequantitiethathe  will  haue  one  ofrbe 
vnequall  fydes  to  be,  andfo  draw  alittle  part  of  the  circle,sc 
then  extend  it  according  to  the  quantum  that  he  wi!  haue  the 
other  vnequal  fyde  to  be, and  draw  likewyfe'a  little  part  of  the 
circle,  and  that  done,  from  the  point  ofthefedion  draw  the  A  3 
lines  to  the  endes  oftfreline  geuen,as  in  the  figure  here  put.Notethat  in  this  the7 
two  fides  mud  be  Inch, that  the  circles  deferibed  according  to  their  quatitie  may- 
cut  the  one  the  other. 

i^FMficpnd  TroMeme.  cFhefecond  Tmpofition* 

Fro  a  point  geuen  Jo  draw  a  right  Ime  equal  to  a  rightline  geuen* 

ofe  tbat  the  point  geue  be  A let  the  right  linege • 
uenhe'B  CJt 
g  required  fro 
[the  point  A}to 
\drdwe  a  right 
lyne  e quail  to 

_ __  J  he  line  BC\  * 

(Draft  (by  the  fir  ft  pettcio)  from  the 
point  A  to  the  poynte  B  a  right  line  A 
B:  and  Vpon  the  line  A  B  defer ibe(by 
the  fir  ft  propoftt'to )  an  equilater  tri * 
angle  ^and  let  the  fame  be  D  A  B^and 
extedjby  thefecond peticio ythe right 
lines  D  A  &  DB^  to  the poyntes  E 

•£>•*/.  and  i] 


Hew  to  definite 
an  equilater  tri 
angle  redilj 
met  bank allj* 


How  to  deferibe 
an  Ifofceles  tri* 
angle  redilj , 


Hew  to  deferibe 
a  Scalenum  trs-* 
angle  rtdtlj. 


Ctnflruttktt,  ■ 


TPgmenjtratiom 


Two  thiriges  ge- 
'sten  in  this  pro- 

tfcsin 
this  prspoftion. 


portion 
rower  c 


The  fir  ft  cafe. 


The fecond  cafe. 


The  third  cafe. 


TbefirB  cBoo%e 

mdt\(s'(hy  the  third  petkio)making  thecentre  (Band  the/pace  B  Cdefiribea 
circle  CG  H:&  agatne(  by  the  fame  /making  the  centre  !D  and  the fpace  3)  G 
defcribea  circle  G  h\Lt  And  fora/* 
much  as  the  pointe  B  is  the  centre  of 
the  circle C G  H,therfore(by  the  i>, 
definitio)  the  line  BC  is  equal  to  the 
line  B  G'.and  forafmuch  as  the poynt 
3)  is  the  centre  of  the  circle  G  KjL: 
therefore  (by  the  fame)the  line  3)  L 
is  equall  to  the  line  3)  G:of  'tohicbtke 
line  3)  A  is  equall  to  a  line  3)  B( by 
the  propofitio  going  beforefi'rherfore 
the  rejiducyna m elyfhe  line  A  L  is  e* 
qual  to  the  refidue, namely  }to  the  line 
B  G(by  the  third  common  fentence) 

And  itisproucd  that  the  line  B  C  is  e* 
quail  to  the  line  B  GVlf her  fore  eyther  of thefe  lines  ALtr  BCis  equal  to  the 
line  B  GjBut  things  which  are  equall  to  one  and  the  fame  thing  are  alfo  equall 
the  one  to  the  other  (by  the  fir  ft  commo  jentence.)VVherfore  the  line  A  L  is  e* 
qual  to  the  line  B  C.fi /her fore  from  the  poynt  geue  f tamely  yA/s  drawn  a  right 
line  A  L  equall  to  the  right  linegeuen  B  C;  which  l?as  required  to  be  done . 

f  ^  v,  .  J  '  -  *  .  )  ,  *  f  #  '■  r;  .  :  f;  i  - 

,  OfProblemes  and  Theoremes,as  we  haue  before  noted,fome  haue  no  cafes  at  all* 
which  are  thofe  which  haue  onely  one  pofition  and  conftruction:and  other  fome haue 
many  and  diners  cafes:which  are  fuch  propofitions  which  haue  diuers  deferiptions  8c 
coniirudions,and  chaunge  their  pofitions .  Of  which  forte  is  this  fecond  propofition, 
whichis  alfo  aProbleme.Thispropofitionhath  two  thin  ges  geuen  :Natnely,a  pointe, 
and  a  line;  the  thing  required  is,that  from  the  pointe  geuen  wherefoeuer  it  be  put,  be 
d raven  a  ine  equall  to  the  line  geuen. Now  this  poynt  geuen  may  haue  diuers  pofitios 
For  it  may  be  placed  eyther  without  the  right  line  geuen3or  in  fome  point  in  it.  If  it  be 
without  it,  either  itis  on  the  fide  of  it,fo  that  the  right  line  drawen  from  it  to  the  ende 
of  the  right  line  geuen  maketh  an  angletor  els  it  is  put  directly  vnto  it,  Co  that  the  right 
line  geuen  being  produced  lhallfall  vpon  the  point  geuen  which  is  without,Butif  it  be 
in  the  line  geuen,theneitherit  is  in  one  of  the  endes  orextreames  thereof :  or  in  fome 
place  betwene  the  extremes.So  are  there  foure  diuer  s  pofitions  of  the  poynt  in  relpedfc 
of  the  line.  Wher  upon  follow  diuers  delineations  and  conltruftions,  and  confequent* 
iy  varietie  of  cafes. 

For  the  firfl;  cafe  the  figure  before  put,feructh. 

To  the  fecond  cafe  the  figure  hereon  the 
fidefetbelongeth.  And  as  touching  the  or¬ 
der  both  of  conftru&ion  and  of  demonftra- 
tion  it  is  all  one  with  thefirft. 


The  third  cafe  is  eafieft  of  all,name!y,whe 
the  poynt  geuen  is  in  one  of  the  extreames. 

As  for  exaple.,ifit  were  in  the  point  C,  which 

is 


o/Eudides  Elementes  *  Fol.n. 


is  one  of  the  extreames  of  the  line  B  C .  Then 
making  the  centre  the  poynt  C,and  the  fpace 
C£  defcribeacirclei?  L  G\ and  from  the  cen¬ 
tre  C  drawe  a  line  vnto  the  circumference, 
which  let  the  CZ,,which  by  the  definition  of 
acircle^lhalbe  equall  to  the  line  geuen  B  C. 

The  fourth  cafe  as  touching  conftru&ion 
herein  differeth  from  the  two  firfte  ,  for  that 
whereas  in  the  you  are  willed  to  draw  a  right 
line  from  the  poynt  geuen,namely ,  A, to  the 
poynt  B  which  is  one  of  the  endes  of  the  line 
geuShereyou  fhal  not  nede  to  draw  that  line, 
for  that  it  is  already  drawemAs  touching  the 
reft,both  in  conftru&ion  and  demonftration 
you  may  proceede  as  in  the  two  firfte.  As  it  is 
manifefte  to  fee  in  thys  figure  here  on  the  fide 
put. 

This;  propofition  for  the  playnes  &  cafi- 
nes  thcreofifeemeth  to  be  as  it  were  a  princi¬ 
ple, and  may  eafly  mechanically  be  done.  For 
opening  the  compafle  to  the  quantitve  of  the 
line  geuen ,  and  fetting  on  foote  of  it  fixed  in 
the  poynt  geuen  and  marki  ng  with  the  other 
another  poynt  wherfoeueritfall,  &  fo  by  the 
firft  peticion  drawing  a  right  line  fro  the  one 
of  thofe  poyntes  to  the  othcr,the  fayd  righte 
line  fhall  beequall  to  the  right  line  geuen :  yet 
in  deede  is  it  no  principle,  for  that  it  may  by 
demonftration  be  proued:  but  principles  can 
not  be  proued,as  we  haue  before  declared. 


5^  The  3,  Trobleme.  T  he  \fPropofitm. 

T wo  unequal  right  lines  beinggeuenjto  cut  of  from  thegrea* 

ter,a  right  lyne  equall  to  the  lefle. 

\Vppofe  that  the  tipo  Unequal  right  linesgeuen  be  AS  O' 
Cy  ofiphich  let  the  lyne  AS  be  the  greater.  It  is  requi * 
red  from  the  line  AS  beinv  the  greater  yto  cut  of  a  right 
line  equal  to  the  right  line  Cyphic-h  is  the  lejle  HneAralPf 
m(hy  the Jecond  propofition) fro  the  point  A  a  right  line 
equall  to  the  line  Cyand  let  the  fame  be  A  D:and  making 

_ \the  centre  Aland  the  fpace  AD  deferibe  (by  the  third 

peticion)  a  circle  DEF.  And  forafmuche  as  the 
point  A  is  the  centre  ofy  circle  ID  E  F,  therfore  A 
E  is  equal  to  A  Dfut  the  line  C  is  equal  to  the  line 
A  DyVherfore  either  of  the fe  lines  A  E  andC  is 
equall  to  A  Dy  therfore  the  line  A  E  is  equall  to 
the  line  Cypher  fore  tlvo  Pne  quail  right  lines  being 
geuen  gamely,  A  S  and  Cohere  is  cut  of  from  AS 
being) greater,  a  right  line  A  E  equall  to  thelejfe  b/ 
ljne3  namely %  to  C:  yphicb  Spas  required  to  be  done.  aih\  This 


) 


The  fourth  tafe. 


T his  prspeftian 
though  it  be 
rj  eafte  to  be 
done  mechanic 
tally, jet  isn« 
principle. 


/ 


Sfjpo  tbinges  ge- 
teen  tn  this  pro- 
paftion. 

Diners  cafes  in 

it. 

viW  v-.vVC 


The  frfl  cafe. 

T he feeond  cafe. 


Th  e  third  cafe „ 


"TheJirHTSoo^e 


This  pro.pofitionpwhich is  a  Probleme,hath  two  thinges  getter),  namely, rare* 
vnequali  right  lines:  the  thing  required  isyfrorrithe  greater  to  cut  of 'a  line  equal 
to  the  idle.  It  hath  alio  diuerscafes.FoTtheliites  geuen  either  are  diflin&th’onc 
from  the  other:  or  are  ioyned  together  at  orieb'ftheir  endes:  or  they  cut.  the  one' 
the  otber,orthe  onecutteth  the  other  in  one  of  the  exxreames.  V  Vhich  may  be 
two  way  es.  For  ether  the  greater  cutteth  the  lefTejOr  the  ieifethe  greater.If  they 
cut  the  one  the  other, eitherech  cuttcth  th’other  into  equali  partes  ?  or  into  vne- 
quall  partes  :  or  the  one  into  equali  parses  ,and  the  other  into  vnequali  partes. 
V  V’hieh  may  happen  i n  two  forts*  for  the  greater  may  be  cut  into  equali  partes,, 
and  the  idle into  vnequali  partes:  or  contrariwife* 

j  *  ‘  ,  TO  \>  07  v  .  lor  V’  ■  .• 

Wh  en  the  vnequali  lines  geuen  are  diftind  the  0ne  hom  the  other ,  the  figure  be¬ 
fore  put  ferueth. 

If  they  be  ioyned  together  at  one  of  their  ‘ 
ends,itis  eafieto  do. For  making  the  centre  that' 
end  where  they  are  ioyned  together ,  &  the  fpace 
the  le{feline,defcribe  a  circle:  whiche  fhall  of  ne- 
ceffitiefby  the  definition  of  a  circle )  cut  offiom 
the  greater  line  a  line  eq nail  to  the  lefie  line ,  as  it ' 
is  playne  to  fee  in  the  figure  here  put.  . 

But  if  theonecut  the  other  in  one  of  the  ex-* 
tremes.  As  for  exaple  :  Suppofe  that  the  vnequali 
right  lines  geuen  be  A  B  and  CD, of  which  Jet  the  . 

line  CD  be  the  greater  :  And  let  the  line. CD  cut 
the  line  A  B  in  his  extreameC.  Then  making  the 
centre  A  and  the  fpace  A  i?,defcribe  a  circle  B  F. . 

An4  vpon  the  line  AC  describe  an  equilater  tri-  , 
anglefby  the  firfljwhich  let  be  A  E  C:8c  produce 
the  lines  E  A  and  E  C,  And  againe  making  the  ce- 
tre  E  and  the.fgacevE  £  defcribe  a  circle  G  £.I,ike«  -  1 
wife  making  the  centre  C and  the  fpace  C<j,def-  ' 
cpibe  a  circle  G'  Z-.Now  forafmuch  .as  theJine'A  F,  , 
is  eq  hah  to  the  litre  £  G  (For  Eis  the-  centre)  of  f 
which  the  line  EA  is  equali  to  the  iine£'C  :  ther 
fore  the  refidue  A  F  is  equals  to  the'  refidue  C  G. 

Bps. the  line ist equali  tothq.  line  ABfov  A 
is  the  centre, wherefore  alfo  the  line  CCj  is  equal) 
tothe1me.i4£.Butthe  line  C  G  is  alfo  eqn'all  to 
theline  C  L^ot  the  point  Cis  the  centre. Where-  . 


The f earth  cafe 

The fifth  cafe* 
The  f  set  cafe. 


But  now  let  CD  be  IefTe  then  A  B,znd  let  it  cut  A  B, 
y  his  extreameC !  Now  then  cythe-rit  cuttein  ltiq  the  - 
in  iddeft  or  not  in  the  middelt.Frrft  let  it  cut  itin  the  niid 
deibthen CDis ether  the halfe q£A,B& fo is ./fCequal 
to  CD. 


Or  it  is  leffe  then  the  halfe :  and  then,  mahiqg  th? 
centre  C&  the  fpace  C  D  defcribe  a  cH!Fe?eiwhich  fhTll;  cut1 
of  from  the  linear/  B  a  line  equ^t.b^j^be'.lifte.C^.nHsi':  r .  ■ 
Or  it  is  greater  then  the  half.  And  the  vntp  the  point  J 
A  put  the  line  A  F  equali  to  the  line  CDyby  thbiiecotid.  ’  ’ 
And  making  the  centre  A  &  the  fpace  A  F  deferibe  a  cir¬ 
cle,  which  ihallcutoffrom  the  line  A  B  a  line  equal!  to 
the  line -^F.thatisvvnto  the  line  CD,  T  £u£ , 


i<n 


"t 


0, 


Fol.n. 


But  if  the  line  C  D  do  not  cut  the  line  AB  in 
the  midft:  C  D  fell  either  be  the  hake  of  the  line 
J.B\ or  greater  then  the  halfe,  or  leffe .  If  C  D  be 
the  halfe  of  A  B,  or  leffe  then  the  half  of  A  2,the 
making  the  centre  C,and  the  fpace  C  D  deferibe  A 

a  circle  whiche  fhall  cut  of  from  the  VmeABa. 
line  equal  to  the  line  CD. 

But  if  it  be  greater  then  the  halfe,then  againe 
vnto  the  point  tA  put  the  line  A  F  equal  to  the 
line  C  ©  (by  the  fecond  propofitio:  )  &  making 
the  centre  A,  and  the  fpace  A  F  defer ibe  a  ctr  chr 
which  {hall  cut  of  from  the  line  AB  a  line  equal! 
to  the  line  iA  F, that  is,to  the  line  C  D . 

But  if  they  cut  the  one  the  other  as  the  lines  C 
D8zAB  do.The  making  the  cetre  B  &  the  fpace 
B  A  deferibe  a  circle  A  F,&  draw  a  line  from  the 
point  B  to  the  point  C,&  produce  it  to  thepoint 
F.And  forafmuch  as  the  two  right  lines  B  F  and 
CD  are  vnequalband the  lined)  cutteththc 
line  BF  by  one  of  his  extreames,  therefore  it  is  ^ 
poffibleto  cut  of  fromCD  aline  equall  to  the 
line  BF. For  how  to  do  it  we  haue  before  decla- 
red,wherefore  it  is  pofllble  from  the  line  C  Dto 
cutofaline  equall  totheline  AB:o:AB  and 
BF a,te equall  the  one  to  the  other. 


ftrufhon  and  demonirracion,proccde  as  in  the  hr  it  cale.  ror  it  1?  ^ 

pofition  to  put  to  the  ende  of  the  greater  lyne  a  line  equal!  to  the  leffe  lvne ,  and 
fo  maky  ng  the  centre  the  fay  d  eude,and  the  fpace  the  leffe  linc,to  deferibe  a  cir* 
cle, which  iliall  cut  of  fro  tOc  greater  lyne  a. lyne  equall  to  the  line  put ,  namely, 
to  the  leffe  line  geuen,as  it  is  manifeft  to  feei'h  the  figures  partly  fiere  vnder  fee. 


The feuentk  QT 
eight  cafes. 


The  ninth  exfi. 


The  tenth  e*fe. 


InnUthefee** 
fes  the  conflru* 
it  ton  end,  demo* 
ft  ration  of  the 
frfl  cafe  wiH 
feme t 


-1  A  4 

: .  tt 

/ 

7 

\  \ 

l-  -  _« 

•5.  '  .  «  .  i  i 

/:  ■ 

'  ’  > 

.v. 

* * 

\ 

-  N, 

'  •  V  - 

'  /  -• ;  v.;  - 

i  *  • 

o 

D.iiii. 


B 


If 


I 


.5\v.1  Jv.Vil  v'.  r; 


TUs PropoJ!-  man  mc- 

tio»,thoHghe  chanically  and  redis 
aifoit  ie  ly  do  this  propofm* 

•/noli  eafie  t*  1  r  r  t 

is  done me-  on>  noc  regardyng 
ehanicatly.  demonftration,  hec 
jet  it  noprin  may  extende  his  co- 
€i*le'  paffe  accordyng  to 
the  quantitie  of 
lefTclyne  geucn, 
fo  feton  foote  there¬ 
of  in  ohe  of  the  ends 
of  the  greater  lyne 
geuen,andwith  the 
other  foote  marke  a  pointe  in  the  faid  greater  lineywhich  fhall  ctittcof  from  the 
greater  line  a  line  equaU  to  the  lefle.The  eafines .of  doing  wherof  may  caufe  this 
m-opofition  alfo  to  feeme  vnto  fome  to  berathera  principlesthena  proportion. 


But  to  that  we  haue  before  in  the  former  propoiitionaunfwered. 


\  . ■:  • .  *,  v\  ^  ^ 

^vbv-..i  ,  ,\  <  i  rj  u 


>  v:.  rr,j  -; 


t  ^  r,  c  ri  •* r 
%  .  l  J  -  i  j  .  * 


i  j  y  ji  qi  ax]  j  l  • 


(W 

5  hv'li  f .  ,  ; 


03  v 


tJt 
'iiOCJ 


{ 

\ 


i .  SvWgr[l  Theorem.  The  \fPropfimn. 

ff there  be  two  triangles  ,of which  twojides  oftldone  he  equal 
to  twojides  of  the  other, eche  fide  to  his  correfpondentfide,and 
loaning  aljo  on  angle  of  the  one  equal  to  one  angle  of  the  other , 
namely,  that  angle  which  iscontayned ynder  the  equall  right 
lines :  the  bafe  aljo  of  the  one  fhall  be  equall  to  the  baje  of  the 
other, and  the  one  triangle fhall  be  equal  to  the  other  triangle-, 
and  the  other  angles  re/nayningfhal  be  equall  to  the  other  an~ 
gles  remayning,the  one  to  the  other,  vnder  which  are fubt en¬ 
ded  equall fides . 

.iiiKCt  c  , 

^  ouffofi 


ofEiitlides  Elemefiteh 


FoLq. 


Vppofetbat  there  be  two  triangles  A  B  C&  ©  E  Fyhd* 
uing  t  wo  fides  of  the  one, namely  A  Byand  A  C,  equall to 
two  fides  of  the  other , namely ,to  ©  E  and  T)Fythe  one  to 
the  other, that  isyA  Bto(D  E,andACto  1 'DFibauyng  al * 
.jo  the  angle  B  A  C, equall  to  the  angle  £©F.  Then  I fay 
| that  the  ha/e  alfo  BC  is  equall  toy  bafe  E  F:  &j  the  tru 
|pj  jangle  A  B  Cjs  equall  to  the  triangle  ©  E  F:andy  the  o * 

'  tber  angles  rematnyng  are  equall  to  the  other  angles  re* 

mayningfbconeto  the  other  yonder  which  are  fubtended  equall  fyder.tb  at  is,y 
the  angle  ABC  is  equall  to  the  angle  ©  E  F,  andj  the  angle  ACB  is  equall  to 
to  the  angle  ©  £  E.For  the  triangle  A  B  C  ex* 
aclly  agreyng  with  the  triangle  ©  £F,  and  the 
point  A  being  put  Vpo  the  point  TEpr  the  right 
Ime  A  B  Vpon  the  right  line  ©  Ey  the  point e  B 
alfo  pall  exaBly  agree  'With  the  pointe  E:for 
float  ( by  fuppofitionjtbe  line  A  B  is  equal  to  the 
line  ©  E.  Jnd  the  line  A  B  exaBly  agreeyng 
• with  the  line  ©  Eythe  right  line  alfo  A  C  exaB* 
ly  agreeth  'With  the  right  line  T>  F,  for  that(by 
Juppofitionjthe  angle  B  A  Cis  equall  to  the  an* 

gle  £©  F .And  for  a] much  as  the  right  line  A  Cisqlfofby  j uppofition )  equall  to 
the  right  line  T>F,  i  her  fore  the  point  e  C  exaBly  agreeth  With  the  pointe  F.  A* 
game  forafmuch  as  the  pointe  C  exaBly  agreeth  with  the  poynte  F,  and  the 
point  B  exaBly  agreeth  with  the  point  E :  therefore  the  bafe  B  CjJoall  txaBly 
agree  with  the  bafe  E  F.For  if  the  point  S  do  exaBly  agree  With  the  point  £, 
and  the  point  C  with  the  point  Fyand  the  bafe  B  C.  do  not  exaBly  agre  Wytb  the 
bafe  E  F,  then  two  right  lines  do  include  a  fuperfcies:  which  (by  the  io.  com  on 
Jentencefis  impoffbk.VV  berfore  the  bafe  B  C  exaBly  agreeth  the  bafe  E  F, 
and therforeis  equallvnto  it.  VFherfore  thewhole  triangle  ABC  exaBly  a* 
greet h  with  the  whole  triangle  ©  EFy<zsr  tberfore(by  the  8,  common  Jentence ) 
is  equall  Vnto  it.  And  (by  the  fame)  the  other  angles  remay  ning  exaBly  agree 
With  the  other  angles  remay  ningyind  are  equall  the  one  to  the  other,  that  is, the 
angle  A  BC  to  the  Angle  ©  E  Fy  and  the  angle  ACB  to  the  angle  ©  F  £t  If 
therfore  there  be  two  triangles, of  which  two  fides  of  the  oney  he  equall  to  two 
fydes  of  the  other, eche  to  his  correfpondent  fide, and  hailing  alfo  one  angle  of  the 
one  equall  to  one  angle  of  the  other }  namely, that  angle  which  is  ckntayntd  y>n* 
der  the. equall  right  lines:  the  bafe  alfo  of  the  one  pall  be  equall  to  the  bafe  of  the 
other, and  the  one  triangle  Jhall  be  equal  to  the  other  triangle, and  the  other  an* 
gles  remainyng  f hall  be  equall  to  the  other  angles  remay  mngy  the  one  to  the  a* 
ther  fender  which  are fubtended  equall  fydes:  whiche  thing  was  required  to  be 
demonfirated. 

This  Proportion  which  is  aThcorerae^hath  two  things  geuen,*  namely,the 

.  E.i.  equality 


Demonstration, 
leading  io  an 
abfirditie. 


T tvs  t  htngts 
»en  in  this  post 
go  ft  son. 


r&w  thingn  e<laa^ty  of  two  fides  of  the  one  triangle,  to  two  fides  of  the  other  triangle^  and 

retired  in  it .  the  equaiitie  of  two  angles  contayncd  vnder  the  equall  fydes,  In  it  alfo  arc  thre 

thinges  required.Thc  equality  ofbafe  to  bafe:  the  equality  of  field  to  field:  and 
the  equality  of  the  other  angles  of  the  one  triangle  to  the  other  angles  of  theo* 
ther  triangle,  vnder  which  are  fubtended  equall  fides, 

*  •  '  . 

Hove  one  fide  it  r\  r  r  r  t  r 

equniito  an  t-  u ne  Ime  ol  a  play  nc  figure  is  equall  to  an  other,  and  fo  generally  one  right 

ther,  &  fo  gene  iync  is  equall to  an  other ,  when  the  one  being  applied  to  the  other,  theyrex- 
r*fitfahe-  £rearnes  agree  together .  For  otherwife  euery  righte  line  applied  to  any  right 
dtTJno-  lyne,agreech  therwith:but  equall  right  lines  only,agree  in  the  extremes. 

ther, 

Hom one reniih  ^nc  re'^rl^nc^  angle  is  equall  to  an  other  reftilined  angle,  when  one  of  the 

wed  angle  is  e~ "  fides  which  comprehendeth  theoneangle,beingfetvpon  oneof  the  Tides  which 
qu*t  to  an  other  comprehendeth  the  ocher  angle, the  other  fide  of  the  oneagreeth  with  the  other 
fyde  ofthe  other.  And  that  angle  is  thegreatcr,whofe  fyde  falleth  without:  and 
that  the  lefle, whole  fydefalieth  within. 


Whj  this  parti* 
ele,ech  to  hts 
correfpondent 
fidefs  put. 


Horn  one  triaH  - 
gtc  is  equall  to 
-an  other . 

What  the  fields 
er  area  of  a  trt- 
angle  ss  , and  fo 
vfanjrcBilined 
figure. 

What  the  cir¬ 
cuits  or  copajfe 
af  a  triangle  is, 
and (o  alfo  of  a- 
nj  reHilmed  fi¬ 
gure. 


Where  as  in  this  propofition  is  put  this  particle  echetohis  correfpondent  fide,  (ia 
ftede  wherof  often  times  afterward  is  vfed  this  phrafe^  one  to  the  other  jit  is  ofne- 
ceffity  fo  put.  For  otherwife  two  fydes  ofone  triangle  added  together,may  bee- 
quail  to  two  fydes  ofan  other  triangleadded  together,5 and  the" angles  allocon- 
tayned  vnder  the  equall  fydes  may  be  equall:  and  yetthe  two  triangles  may  not- 
withltandingbe  vnequall*  Where-note  that  a  triangle  is  faydtobe  equall  toaa 
other  triangle, when  the  field  or  area  of  rheone  is  equall  to  the  areaof  the  others 
And  the  area  of  a  triangle, is  that  fpace, which  iscontayned  within  the  fydes  ofa 
triangle,  Andchecircuiteorcompafleofatriangleisalinccompofed  of  all  the 
fides  ofa  triangle. And  fo  may  you  think  ofali  other  reftilined  figures.  And  now 
to  proue  that  there  may  be  two  triangles, two  fydes  ofone  of  which  being  added 
together, may  be  equall  to  two  fy  des  ofthe  other  £>  „ded  together, and  the  angles 
contained  vnderthe  equall  fydes  may  be  equall,  and  yet  notwithftanding  the 
two  triangles  vnequall,  Suppofe  that  there  be  two  rcftangle  triangles:  namely, 
A  B  C,andD  £  Fvandlet  their  right  angles  be  B  A  CandE  D  F.  And  in  the  tri¬ 
angle  A  B  C  let  the  fyde  A  B  beg.  and  the  fyde  A  C  4,  which  both  added  toge¬ 
ther  make  7. 

And  in  the 
triagle  D  E 
F,let  the  fide 
DEbe2.and 
the  fide  D  F 
foe  5*  whichc 
added  toge¬ 
ther  make  al 
fo  7*8do  the  B 

fy  des  ofthe  one  triangle  added  together,are  equall  to  the  fides  of  the  other  ?m# 
pdeadded  together, Y etare  both  the  triangles  vnequall,and  alfo  their  bafts.  For 
the  area  ofthe  triangle  A  B  C  is  6  and  his  bafe  is  5.  And  the  area  ofthe  triangle 
D  £  F  is  V-  and  his  bafettfp  2.9.  So  that  to  haue  the  areas  oft  wo  triangles  co  be  c- 
quali,  it  is  requifite  that  all  the  fydes  of  the  two  mangles  be  equall,  eche  tohys 
correfpondent  fyde. It  happenethalfo  fometymes  in  triangles,  that  the  areas  of 
them  bey  ugcqua.ll,  their  fydes  added  together  fhall  be  vnequall.  Andconccan- 


ofEuclides  Elements. 


Foil fa 


wife^hcir  Tides  beyng  equall,thrir  areas  be  vnequalUs  in  theCe  figures  hcrepuc 
it  is  plaine  to  fee*  In  the  firft  *  ^ 

example  the  areas  of  the  two 
triangles  arc  equal, for  they  are 
eche  i2»and  the  fides  inech  ad¬ 
ded  together  are  vnequall,  for 
in  the  one  triangle  the  fides  ad¬ 
ded  together  make  18.  and  in 
the  other  they  make  16.  But  in 
the  fccondexaple  the  areas  of 
the  two  triagles  are  vnequal, 
for  the  one  is  i2*and  th’other 
is  l§*Butthe  fides  added  tos 
gethcr  in  eche  are  equall,  for 
in  eche  they  makeiS. 

That  angle  is  faid  to  fub- 
tend  a  fide  ofa  triagle, which 
is  placed  dire&ly  oppofite, 

8c  againfi:  that  fide.That  fide 
al  fo  is  fay  d  to  fubtend  an  an¬ 
gle, which  is  oppofite  to  the  angle. For  eucry  angle  ofa  triangle  is  contaynedof 
two  fy  des  of  the  triangle,and  is  fubtended  to  the  third  fide* 

This  is  thefirfi:  Propofition  in  which  is  vfed a  demonft  ration  leading  to  an  Thisfrofofatien 
abfurdicie,oran  impoffibilitie,Vyhich  is  a  demonftration  that  proueth  not  di-  tr,uefjky* 
redly  the  thing  emended,  by  principles,  or  by  thinges  before  proued  by  thefe  hZgtolTaifar 
principles:but  proueth  thecontrary  therof  to  be  impoffiblc,&:  fo  doth  indirect-  dity, 
iy  proue  the  thing  emended, 


A*  % 

AL — t — 4 — 

5  X 

— , — i 

* 


Hti®  an  angle 
is  fayd  to  fainted 
a  fade  sand  a  fade 
an  angle. 


\  i. 


k 

ty&The  i.T  heoreme.  Tbej.Trofo/ttion. 

(tJn  lfofcelesjr  triangle  of two equal ' fide sjhath  his  angles  at 
the  bafe  equall  the  one  to  the  other.  <iAnd  thofe  equal  fides  be- 
ing produced jhe  angles  which  are  Under  the  bafe  are  alfo  e* 
quail  the  one  to  the  other. 

. 


*  ^  • 

mVppofetbdt  ABC  lea  triangle  ‘of  tn>o 

equall  fy  des  called  Ifofcelesy  hauing  the 
!  [fade  A  B  equall  to  the  fide  A  C .  And 
}  (by  the  fecond peticio)produce  the  lines 
A 'Bo*  AC  directly  toy  points  (DoEpThe  I  fay, 
that  y  angle  ABC  is  equal  to  the  angle  ACB:andy 
ji  angle  C  B  2)  is  equal  to  tj?  angle  B  C£,Tdke  in 
the  line  B£)a  point  at  all  aduentures,  and  let  the 

fame 


J  w  *  V 


/ 


fTbejirBcBookg 

fame  be  F }and(by  the  third propofttion)from  the  greater  line,  namely %  AE9 
cut  of  a  line  equall  to  A  F  being  the  lefle  lineyand 
let  the  fame  be  A  G :  and  draw  a  right  line  fro  the 

pfmt  F t0  l!je  Pomt  C>af1^  an  0t^er from  the  point 

G  to  the  point  BJN.OW  then  for  as  muche  as  AF  is 
equall to  A  G^and  A  B  is  equall  to  A  C,  therefore 
thefe  tWo  lines  F  A  and  A  C  are  equall  to  thefe  two 
lines  G  A  and  ABfthe  one  to  the  other ,  and  they 
containe  a  common  angle, n am ely,that  which  is  co* 
tained  Vnder  FA  G:  w  her fore  (by  the  fourth  pro » 
poftion)the  bafe  F  C  is  equall  to  the  bafe  G  <B:  and 
the  triangle  A  F  C  is  equall  to  the  triangle  A  G  Byand  the  other  angles  rental* 
ning.are  equall  to  the  other  angles  remaining  the  one  to  the  other  yVnder  which 
are  fubtended  equall fdes :  that  is,tbe  angle  ACFis  equall  to  the  angle  ABG , 
and  the  angle  AFC  is  equall  to  the  angle  A  G  (B,  And  for  af much  as  the  whole 
line  A  F is  equall  to  the  whole  line  A  Gyof  which  the  line  A ‘B  is  equal  toy  lyne 
A  C,  tberfore  the  ref  due  of the  line  A  F ^namely  ythe  line  B  Ffs  equal  to  the  re * 
fidueofthe  line  AG ^namely, to  the  UneCG  (by  the  third  common  fentence ) 
And  it  is  proued  that  C  Fis  equal  to  B  G .  Not*  therfore  thefe  tWo  BF&FC 
are  equall  to  thefe  two  CG  and  G  B  the  one  to  the  other,  and  the  angle  B  FCis 
equall  to  the  angle  C  G  B,and  they  haue  one  bafe, namely, B  Cycommon  to  them 
both:  w  her fore  (by  the  4.  propofttion)  the  triangle  B  FCis  equall  to  tin  trU 
angle  CGBy  and  the  other  angles  remaynyng  are  equall  to  the  other  an* 
gles  remaining  eche  to  other y  Vnder  which  are  jubt ended  equall  fides.Wher* 
fore  the  angle  F  BC  is  equall  to  the  angle  GCBy  and  t  he  angle  B  C  Fis  equall 
to  the  angle  CBG .  ISloW  forafnuch  as  the  whole  angle  ABG  is  equall  to  the 
“whole  angle  AC  F(  as  it  ha  th  bene  protied)of which  the  angle  CBG  is  equal  to 
the  angle  B  CF:  therfore  the  angle  remayning:  namely ,A  B  C  is  equall  to  the 
angle  remaining3namelyyto  A  C  B(by  the  third  common  fentence)  And  they  ar 
the  angles  at  the  bafe  of the  triangle  A  B  C.a  nd  it  is  proued  that  the  angle  FB 
Cis  equall  to  the  angle  G  C  Byand  they  are  angles  Vnder  the  bafe .  W her  fore  a 
triangle  of  two  equall  (1 ides  hath  his  angles  at  the  bafe  equall  the  one  toy  other # 
And  thofe  equall  Jides  being  producedy  the  angles  which  are  Vnder  the  bafe  art 
alfo  equall  the  one  to  the  other :  which  Was  required  to  be prou  ed. 

for  that 


of EuclidesElementes.  Fd.\ 

fides  of  the  triangles  A  FC  and  AGB&  alfo  thefy  des  of  the  triagles  B  F  C  SC 
C  G  B  run  fo  one  withinan  other,thcrforc  I  hauc  here  put  the  diftin&iy,name* 
ly,the  triangles  F  A  C  and  B  t  C  on  one  fyde  of  the  figure  of  the  propofitio  SC 
the  triangles  A  GB  and  C  G  B  on  the  other  fydc.-fo  thatyou  may  with  Idle  la¬ 
bor  fee  the  demonftracionplayncly, 

ir 

That  tn  an  Ifofceles  triangle,thetwo  angles  aboue  the  bafe  arcequall,may 
Cuherwife  be  demonftrated  without  drawing  lines  beneath  the  bafefomwhac  ab 
tering  the  conftru&ion.Namely, drawing  the  lines  within  the  triangle  jwhiche 
before  were  without  it  after  this  manner. 


Suppofethat^-ffCbe  an  Ifofceles  trianglc:andinthclinc 
A  B  take  a  point  at  all  aduentures.and  let  the  fame  be  D ,  And 
from  the  line  AC  cut  of  a  line  eqnalltothe  line  AD.  Which 
let  be  A  £,And  draw  thefe  right  lines  B  E,D  C, and'D  E ,  Now 
forafmuch  as  in  the  triangles,^  B  E, znd  AC  D>  the  fifie  AB  is 
equall  to  the  fide  A  C, by  fuppofition ,  and  the  fides  A  D  and 
A  E  are  alfo  equall  by  conftrudion,and  the  angle  at  the  poynt 
A  is  common  to  them  both :  therfore.by  the  fourth  propofiti- 
on,the  bale  B Eis  equall  to  the  bafe  D  C. And,by  the  fame,  the 
angles  remayningofthe  one  triangle  a  re  equall  to  the  angle  s 
remay  ning  of  the  other  triangle.  Wherefore  the  angle  A  B  Eis 
equal  to  the  angle  A  CD.Againe  forafmuch  as  in  the  triangles 
B  'D  £,and  CED  the  fide  DBis  equall  to  the  fide  £  C,  and  the 
fide  B  E  to  the  fide  D  C, and  the  angle  DB  E  is  equal  to  the  an¬ 
gle  £  C  D, and  the  bafe  D  E  being  common  to  both  triangles  is  equall  to  it  felfe: there¬ 
fore  the  angles  remayningofthe  one  triangle,  are  equall  to  the  anjgles  remayningof 
the  other  triangle.  Wherfore  the  angle  ED  Bis  equall  to  the  angle  D  EC:  8c  the  angle 
DEB  is  equal  to  the  angle  £  D  C.And  forafmuch  as  the  angle  E  D  B  is  equal  to  the  an 
gle  DEC,  fro  which  are  taken  away  equall  angles  D  E  B,&  E  D  C ,  therfore  by  the  co- 
mon  fentence  the  angles  remayning,namely,B  D  C  and  CEB  are  equall;  And  as  it  was 
before manifeft  the  fides  B  D  and  D  Care  equall  to  the  fides  C E  and  £  B  the  one  to  the 
other,thatis,echtohiscorrefpondentfide  :  and  the  bafe  BCis  common  to  both  the 
triangles :  wherfore  the  angles  remayning  are  equall  to  the  angles  remayning  the  one 
to  the  other,vnder  which  are  fubteded  equall  fides. Wherfore  the  angle  D  B  Cis  equall 
to  the  angle  £  C  B. For  the  line  D  C  fubtendeth  the  angle  D  B  C,and  the  line  £  B  fubte- 
deth  the  angle  £  C2:which  two  lines  are  as  we  haue  before  proued  equall.  Wherfore 
in  an  Ifofccics  triangle,the  angles  at  the  bafe  are  equall  ,  though  the  right  lines  be  not 
produced. 

To  prouethis  alfo, there  is  an  other  dcmonflration  of  Pappus  much  fhor* 
ter  which  needeth  no  kindofaddition  ofany  thing  at  all:as  followeth. 

Suppofe  that  ABC  be  an  Ifofceles  triangle  ,&  let  the  fide 
A  B  be  equall  to  the  fide  zA  C.Now  then  vnderftand  this  one 
triangle  to  be  as  it  were  two  triangles.  And  thus  reafon  •  For¬ 
afmuch  as  in  the  two  triangles  A  B  C and  A  CB,<tA B  is  equal 
to  AC  &  AC  to  A  £,therfore  two  fides  of  the  one  are  equall 
to  two  fides  of  the  other,  ech  to  his  correfpondentfide,  &  the 
angle  Cis  equall  to  the  angle  C./££,  for  it  is  one  and  the 

felfefameangle.Wherforeby  the  4.propofition  the  bafe  CB 
is  equal!  to  the  bafe  B  C ,  and  the  triangle  zA B  C  is  equall  to 
the  triangle  A  CB',  and  the  angle  AB  Cis  equall  to  the  angle  q 
ACB, and  the  angle  AC B  to  the  angle  AB  C.-for  vnder  them  ^ 
are  fubtended  equall  fides  ,namely, the  lines  AB  Sc  A  C.  Wher 
fore  in  an  Ifofceles  triangle,the  angels  at  the  bafe  are  equall. 

E,iH,  Tlx<? 


An  ethtr  dema- 
ft ration  tnuen- 
ted  by  Proelut . 


An  ttkerdemf* 
ftration  inuen- 
tedby  Pnppnt. 


Mtlepus 
•the  invent  or  of 
thit propofition. 


Demonffration 
leading  to  an 
dmpojfibthtj^ 


¥:6i  thief  efi  and 
tn  ift  proper  kind 
tsfczmterfon., 

.\c.-wp  \  ,■  %  s'« 


TbefirJb'Bookg 

The  old  Philofophcr  Thales  Milefius  was  thefirft  inuenrer  of  this  fifth  propofiti- 
on,as  alfo  of  many  other. 

^tefThe  third  Theoreme .  'The fixtTmpoJitwn. 

If  a  triangle  haue  two  angles  equall  the  one  to  the  other :  the 
fdes  alfo  of the famejtohich fubtend  the  equall  angles 7(halbe 
equall  the  one  to  the  other . 


ppofe  that  ABC  he  a  triangle ,  hauing  the  angle  ABC 
equall  to  the  angle  AC  3.  Thenlfay  tbattbejUe  AB 


i  ^  f  —  "  *  /  7  -  J - 

I h  equall  to  the  fide  A  C.For  if  the  fide  A  B  he  not  equal 
\to  tbejide  A  C3tben  one  of  them  is  greater  .Let  A  3  he 
■the greater,  And  by  the  third propofticn  Ofom  A  B  he* 
fing  the  greater  cut  of  a  line  equal  to  the  lefie  line  fvhich 
Cj\h  AC And  let  the  fame  be3)B  tAnd  draTbe  a  line front 


thepoynt  3)  to  the  poynt  C.lSloStf forafmuchas  the 
fde  3)  3  is  equall  to  thefyde  A  Cyand  the  line  3 
C is  common  to  the  both  therefore  thefe tlbofydes 
ID  Band  3C  are  equall  to  thefe  Wo  fydes  AC&> 

CB  tbeonetotbeotber  .Andtbe  angle  3)3  Ch 
h  fuppofytion  equall  to  the  angle  AC  3.  VVher • 
fore( by  t  he  4  proportion  )  the  hafe  3)Ch  equall 
to  the  hafe  A  3  :&(by  the  fame)the  triangle  !DB 
C  is  equall  to  the  triangle  AC  3:namelyy  thelefe 
triangle  Vnto  the greater  triagleywhicb  is  impofi 

hle.W  before  the  fyde  A Bis  not  Unequal  to  the  fide  AC .  VVher  fore  it  he* 
qual.If  ther fore  a  triangle  haue  Wo  angles  equall  the  one  to  the  other:the  fydes 
alfo  of  the  fameffohich  ftihtende  the  equall  angles }  fall  he  equall  the  one  to  the 
other *  spbicb  Tl>as  required  to  he  demon flrated. 


’J'.jL  ( 


In  Geomecrieis  oftentimes  vfcdconucrfionofpropofitions.As this  propo.' 
fition  is  the  conuerfeof  the  propofition  next  before.The  chiefeftandmoftjprOC 
per  kind  ofeonuerfion  is,when  that  which  was  the  thing  fuppofed  in  the  formpr 
propofmon}is  the  conciuilon  oftheconuerfeandfecond  propofition  :  and  con? 
trai-fwife  that  which  was  concluded  in  the  firft:, is  the  thing  fuppofed  in.tHc  fe- 
cond  as  in  the  fifth  propofition  it  was  fuppofed  the  two  fides  ofa  triangle  to  be 
equal, the  thing  concluded  is, that  the  two  angles  at  thebafe  are  equall  Sc  in  this 
propofition, which  is  the  conuerfe  therof  is  fuppofed  that  theangles  at  the  bafe 
be  equalftWhich  in  the  former  propofition  was  the  conclufion*And  the  con- 
ciufion  is,that  the  two  fy  des  fubtending  the  two  angles  are  equall, which  in  the 
former  propofition  was  thefuppofition.This  is  the  chiefeft  kind  of  cpniierfiofi 
^oiforraeansUcirayne* 


0, 


T  jpg  conelufont 
in  the  fifth  pro¬ 
pofition. 

T he  fxt  prgpo- 
ftion  ts  the  con-' 
uerfe  as  tea¬ 
ching  the  ftp 
tonclufon  caefye 
The  conuerje  at 
touching  thefe ■» 
send  cotsclufotti 


jl  wfojy* 

Thereis  an  other  kind  ofconuerfion,bur  not  To  full  a  conuerfion  norfo  per-  Another  hind  of 
kSt  as  the  firft  is*  Which  happeneth  in  compofed  propofitions,that  is,  in  fueh, 
which  haue  mo  luppofttions  then  onejandpafife  from  thefe  fuppofitions  to  one 
conclufion.  In  theconuerfes  of  filch  propofiti6s, you  palfefrom  the  conclufion 
of  the  firft  propofition,  with  one  or  mo  of  the  fuppofitions  of  the  lame:  Sc  con¬ 
clude  fome  other  fuppofition  of  the  felfe  firft  propofition:  of  this  kinde  there 
are  many  in  Euclide.Therofyou  may  hauC  an  example  in  the  S.propofition  be¬ 
ing  the  conuerle  of  the  fourth.Thisconuerfion  is  not  fo  vniforme  as  the  other, 
but  more  diuers  and  vneertaine  according  to  the  multitude  of  the  things  geuen^ 
or  fuppofitions  in  the  propofition. 

Bucbecaufe  in  the  fifth  propofition  there  are  two  conclufions,  the  firft,  that 
the  two  angles  atthebafe  be  equall:  the  fecond^that  the  angles  vnder  thebafe  are 
cquall:  this  is  to  bcnoted,that  this  fixe  propofition  is  the  conuerfe  of  the  fame 
fifth  as  touching  the  firft  conclufion  onely .  You  may  in  like  mancr  make  a  con- 
uerfe  of  the  fame  propofition  touching  thefecond  conclufion  therof.  And  chat 
after  this  maner* 

CV  He  ttyo  fides  of  a  triangle  beyng  produced \if the  angles  vnder  the  bafe  be  ecjualfthe  J aid  triangle 
*  jhatl  be  an  Jfofceles  triangle  Jn  which  propofmo  the  fuppofition  ls.that  the  angles 
vnder  the  bafe  are  cquall :  which  in  the  fifth  propofition  was  the  conclufion:  6c 
the  conclufion  in  this  propofition  is.that  the  two  fides  ofthe  triangle  are  equal, 
which  in  the  fife  propofition  was  the  fuppolition.Butnowfor  proofeof  the  laid 
propofition: 

Suppofe  that  there  be  a  triangle  AB  C,  &  let  the 
fides  s^^andix/C* be  produced  to  the  poyntesU 
and  G ,  and  let  the  angles  vnder  the  bafe  be  cquall, 
namely,the  angles  D  B  C,and  GCB .  Then  I  fay  that 
the  triangle  ABC  is  an  Jfofceles  triangle .  For  take  ii* 
the  line  AD  a  point  which  let  be  E,  And  vnto  the  line 
BE  put  the  line  CF  equallfby  the  3  .propofitio  ^).And 
draw  thefe  lines  E  C,B  F,and  E  F.  Now  forafmuch  as 
BE  is  equall  to  C  F,  and  BC  is  common  to  the  both , 
therfore  thefe  two  lines  B  E&B  C*,are  equall  to  thefe 
two  lines  C  F  and  C  B  the  one  to  th e  other,  &  the  an¬ 
gle  E  B  Cis  equall  to  the  angle  FCF  by  fuppofition. 

Wherfore(by  the  4,propofition )  the  bafe  of  the  one 
is  equall  to  the  bafe  of  the  other,and  the  one  triangle 
is  equal  to  the  other  trian  gle ,  &  the  other  angles  re- 
mayning  are  equal  vnto  the  other  angles  remayning, 
theone  to  the  other,vnder  which  are  fubtended  equall  fides,  Wherfore  the  bafe  ACis 
equall  to  the  bafe  FF,and  the  angle  B  EC  to  the  angle  CEB  ,  and  the  angle  CB  Ft  o 
the  angle  B  C  F.Bqt  the  whole  angle  EBCis  equal!  to  the  whole  angle  F  CB ,  of  which 
the  angle  F  B  Cis  equall  to  the  angle  ECB  :  wherefore  the  angle  remayning  EB  Fisc-* 
quail  to  the  angle  remayning  F  C  F.But  the  line  B  Eis  equall  to  the  line  C  F,Sc  the  line 
B  F  to  the  line  CE} and  they  contayne  equall  angles:  wherfore  by  the  fame  fourth  pro¬ 
pofition  the  angle  B  E  Fis  eq uall  to  the  angle  C F  F.  Wherfore  by  this  fixt  propofition 
the  fide  <iA  Eis  equall  to  the  fide  A  F: of  whiche  B  E  is  equall  to  C  F,  by  conftru&ion : 
wherforefby  the  third  common  fentence) the  refidue  A  B  is  equall  to  the  refidue  A  C 
Wherfore  the  triangle  A  B  Cis  an  Jfofceles  triangle. if  therfore  the  two  fides  of  a  trian¬ 
gle  being  produced, the  angles  vnder  the  bafe  be  equall,the  fayd  triangle  fhall  be  an  /- 
fifieUs  triangle:which  was  required  to  be  proued. 

This  moreouer  is  to  be  noted, that  in  this  propofition  there  may  be  an  other  cafe*  a*  other  cafe  its 
for  in  taking  an  equall  line  to  eA  Cfrom  A  B,  you  may  take  it  from  the  poynte  cA  and  thufxtpropof. 

E»iiij.  not  <  tnn. 


iionJiruiiioffa 

Demtnpiratiotii 


Thefirft'Bookp 


Mvt  from  the  poynt  ,  And  yet  though  this  fuppbfition 
aifo  be  put  the  felfe  fame  abfurdity  will  follow. 

For  fuppofe  that  A  C  be  equall  to  A  D  :aiid  produce 
the  line  CiA  to  the  poynt£:  and  put  the  line  AE  equal! 
to  the  line  TO  Z?(by  the  third  propolitionjwherefore  the 
whole  line  C  E  is  equall  to  the  whole  line  AB  (  by  the  fe- 
cond  common  fentece )  Draw  a  line  from  the  poynt-E  to 
the  point  jB.Andforafnuich  as  the  lineaxf  B  is  equall  to 
the  line  £C,  and  the  line's  Cis  common  to  them  both, 
and  the  angle  <sA  CB  is  fuppofed  to  be  equall  to  the  an¬ 
gle  SC:  Wherfore  (by  the  fourth  propofition )  the  tri¬ 
angle  EBCis  equall  to  the  triangle  AB  C,  namelye,the 
whole  to  the  part:  which  is  impoffible. 


t 


Qtman’lrdt'tcn 
le*dmg  to  An 
abjnriittfe'. 


The  4..  Theoreme.  The  7.  Tropofition - 

fffrom  the  endes  of  one  line,  be  drawn  two  right  lynes  to  any 
pomteithere  can  not  fro  the felffame  endes  on  the fame  fide, be 
drawn  two  other  lines  equal  to  the  two  firjl  lines ,  the  one  to  the 
other gonto  any  other point. 

Or  if  it  be  pofitble:  then  from  the  endes  of one  &  the  felf 
fame  right  line ,  namely, A  <3,  from  thepointesf  I fay)  A 
and  3, let  there  be  drawn  tspo  right  lines  A.  Cand C3to 
the  point  C;  and  from  the  fame  endes  of  the  line  A  3,  let 
there  be  dr  assert  tSbo  other  right  right  lines  A3)  and  3) 
3  equall  to  the  lines  A  Cand  C  3  the  one  to  the  other # 

_ _ _ js,eche  to  his  eorrefpondent  fme^and on  one  and  the fame 

fide, and  to  an  other  point e 3  namely ,  to3):fo  that 
let  C  A  be  equall  to  3)  A  beyng  both  drawen  from 
one  end,that  is,A:&  let  C3be  equall  to  3)  3,be» 
yng  hothalfo  draw  from  one  ende,that  isfi.And 
(by  the  firjl  peticion)  draSP  a  right  line  from  the 
point  Cto  the  point  3).Kow  for aj much  as  A  Cis  e* 
qual  to  A  3),the angle  A  C3)alfo  is(by  the  i.pro* 
pofition)e quail  to  the  angle  A3)C:  Therfore  the 
angle  A  C3)  is  lefie  the  the  angle  3  3)  C,  Wher* 
f ore  the  angle  3  C  3)  is  much  le fie  then  the  angle 
3  3)  C.  Againe  forafmuch  as  3  C  is  equall  to  3  3),  and  tier  fore  dlfo  the  angle 
3  C  D  is  equall  to  the  angle  3  3)  CtAnd  it  is  proued  that  it  is  much  lefie  then  it: 
which  is  im pofitble  Jf tber fore from  the  endes  of  one  line ,  be  drawen  two  right 
lines  to  any  point  et  there  can  not  from  the  felfe  fame  endes  on  the  fame  fide  be 
drawn  ftoo  other  lines  equall  to  the  two firjl  lines ,  the  one  to  the  other  jvnto  a* 
ny  other  point:  Which  ip  as  required  to  be  demonfir  at  ed^  In 


of Eudides  Elements .  FoLij . 

In  this  propofition  the  conclufionisa  negation,  which  very  rarely  happe-  Negative  and* 
neth  in  the  mathematical!  artes ,  For  they  euer  for  the  moft  part  vfc  to  conclude 
affirmaciuely,&:  not  negatiuely .  bora  propofitio  vniuerfallalfirmatine is  moft  uLttartgt!™ 
agreahlc  to  lciences,as  faith  AnDotle^nd  isofit  felfeftrong,  and  nedeth  no  nega- 
due  to  his  proofe.Butan  vniuerfall  propofition  negatiue  muftof  necefiitie  haue 
to  his  proofe  anaffirmatiue,For  of  oncly  negatiue  propofitions  there  canbeno 
deniormrations.Aiid  therfore  fciences  vfingdemonftracion,  conclude  affirraa- 
ciuely^and  very  feldome  vfe  negatiue  conclufions. 

S&dJn  other  demonjlration  after  Campanus . 

Suppofe  that  there  be  a  line  A B ,  from  whofe  ends  A  a ndl?, 
let  there  be  drawen  two  lines  A  C  and  B  C  on  one  fide ,  which  let 
concur  in  the  poynt  C.Then  I  fay  that  on  the  fame  fide  there  can¬ 
not  be  drawen  two  other  lines/rom  the  endesof  the  line -4  2?, 
which  Hull  concur  at  any  other  poynt, fo  that  that  which  is  drawe 
from  the  point  A  fhall  be  equall  to  the  line  A  C,and  that  which  is 
drawen  from  the  point  3  flialbe  equall  to  the  line  BC.  For  if  it  be 
poflible,let  there  be  drawn  two  other  lines  on  the  felfe  fame  fide, 

which  let  concurrent  the  point  D,and  let  the  line  AD  be  equall  to  the  line  ACy8c  the  Dtuers  carts 
line  B  D  equall  to  the  line  B  C.  Wherfore  the  poynt  D  fhall  fall  either  within  the  trian-  thts  gems„j}r^ 
gle  ^  5  C,orwithout,For  it  cannot  fall  in  one  ofhhefides,for  then  a  parte  fiiouldbe  e-  tl0„, 
quail  to  his  whole. If  therfore  it  fall  without:  then  either  one  of  the  lines  A  D  and  D  B 
fhall  cut  one  of  the  lines  A  C  and  Chords  neither  fhall  cut  neyther  .Firfte  let  one  cut  firfieafe. 
the  other  and  draw  a  right  line  from  CtoD.  Now  forafmuch  as  in  the  triangle  ACDr 
thetwofides^4Cand^4DareequaIl,therforetheangle  <•//  CD  is  equall  to  the  angle 
A  DC}by  the  fifth  propofitio:  likewifc  forafmuch  as  inthetriagle5Ci),thetwo  fide* 

2?  Cand  B  D  are  equall,therfore  by  the  fame,  the  angles  BCD  ScBDC  are  alfo  equall. 

An'4  forafmuch  as  the  angle  B  DC  is  greter  the  the  angle  ADC ,  F 

it  followeth  that  the  angle  B  CD  is  greater  then  the  angle  ACD , 
namely  ^the  part  greater  then  the  whole  -.which  is  impolTible. 

But  if  the  point  D  fal  without  the  triangle  ABC  Jo  that  the  lines 
cut  not  the  one  the  other,draw  a  line  from  D  to  C.  And  produce 
the  lines  B  D8c  BC  beyond  the  bafe  CD,  vnto  the  points  E8cF. 

And  forafmuch  as  the  lines  A  C and  AD  2 re  equall,  the  angles  A 
C  D  zndA  D  C  fhall  alfo  be  equall,  by  the  fifth  propofition  :  like- 
wife  for  afmuch  as  the  lines  B  Cand  B  D  are  equal,  the  angles  vn- 
der  the  bafe,namely,the  angles  F  D  C and  E  CD  are  equall ,  by 
the  feconde  part  of  the  fame  propofition  .  And  for  as  much  as  the  angle  ECDis  lefle 
then  the  angle  ACD :  It  followeth  thattheangle  FDCislefie  the  the  angle  ADCi 
which  is  impoffible.’for  that  the  angle  A  DCis  a  part  of  the  angle  F  D  C.  And  the  fame 
inconuenience  will  follow  if  the  poynt  D  fall  within  the  triangle  <lA  B  C, 


c  p  > 


SbfThe,  fift  Theoreme.  "The  8.  Trnpofition* 

Jftxto  triangles  haue  two fdesoftloone  equall  to  two  fides  of 
the  other ,ecbe to  his  correspondent fide haue  alfo  the  bafe 
of  the  one  equall  to  the  bafe  of  the  other :  they  fhall  haue  alfo 
the  angle  contained  vnder  the  equall  right  lines  of  the  one,e-» 
quail  to  the  angle  contayned  vnder  the  equall  right  lynes  of 
the  other.  Fa*  Suppofe 


sv  A 


Demonftrarion 
'teAding  to  an  | 
omfofjlbthtj , 


•>\j 


Tbefir'sl'Booke 

V ppofe  that  there  he  two  triangles  A  B  C and  V  E  F:& 
let  thefe  two  /ides  of  the  one  AB  and  A  CJbe  equal!  to 
theft  two  (ides  of  the  other  ID  E,and  D  Fyechto  his  con 
ref  pendent  fide. that  is ^A  BtoV  E, and  AC  Jo  D  F.Zsr 
'let  the  bafe  of  the  one  jiamely ,B  C  be  equal  to  the  bafe  of 
the  other .namely  jo  E  F.Then  lfay  jbat  the  angle  (BA 
,C  is  e quail  to  the  angle  EDF.For  the  triangle  ABC  ex* 
aftly  agreing  With  the  triangle  DE 
F.and  the- point  B  being  put  Vpon  the 
point  E^and  the  right  line  BCvpon 
the  right  line  B  F :  the  point  C  Jhall 
exactly  agree  la ith  the  point  F  (for 
the  line  B  Cis  equal l to  the  line  EF) 

And  B  C  exaftly  agreeing  With  E  F 
the  lines  alfo  B  A  and  A  C  frail  ex* 
aftly  agree  With  the  lines  E  D&D 
F4  For  if  the  baft  B  C  do  exaftly  a* 
gree  With  the  bafe  F  F.  but  the  (ides 
\B  A  fjr  A  C  doo  not  exaftly  agreed 
the  (ides  E  2)  &  DEJbut  differ  as  F 

G  esrG  Fdce.  the  frornji  endes  of  one  r  j 

dyne  fhalbe  drawn  two  fight  lines  to  a  poynt,&  from  the  felf fame  endes  on  the 
fame  fide  (halbe  drah?h  two  other  lines.equal  to  the  two  fir (l  lines. y  one  to  the  o« 
ther„and  vnto  an  other  poyntibut  that  is  impofttblefby  the  feuenth  propofitio) 
V  F  her  fire  the  bafe  B  C  exaftly  agreeing  with  the  bafe  EFjbe  (ides  alft  BA 
and  AC  do  exaftly  agre  With  the  (ides  ED  and  D  F.W her  fore  alfo  the  angle 
B  .AC Jhall  exaftly  agre  lb  the  angle  E  D  Fyand  therfore  (hall  alfo  be  equal  to 
it. If  the  /  fore  two  triangles  bane  two  (ides  of  the  one  equallto  two  fide  s  of  the 
othenyech  to  his  cor  refpbndent fideyand  haue  alfo  the  baft  off  one  equallto- the 
haft -vf the  other.they  j hall  haue  alfo  theanglecantayitedwdierthe  eqrnll  right 
lines  of the  dne/qUaUto  the  angle  contayhed  Vnder  thefquall  right  linesofthe 
othenwhich  Was  re c 


:  'H*  f\ 


r“>  *r\ 


.  •  i  ; 

Tft i s T heorcme  is  theconuerfe  of tfre  fourth ,  biiti'cisflOtthe  chiefeftand 


,/x 


This  proposition 
is  the  conuer fe 
of  the  fourth, 
but  not  the  chts 
feji  hnd  of  con- 
nerfiott. 


■pofttio  .Whole  conu  erfe  t  h  is  is  ,i  s  a  copound  rheorcm^hariingt^o  tb«%$  gcue 
or  iqpp.o fed,  which  arethefe:  the  one,tharaWq  fides  of  the  one  triagle  be  equal  to 
t  wo Mes  of  the  other  frugie:  th’othefjthat  the  anglecbtaintfd  bf  the  tWo  tides  of 
ttfo^fs'cqUalrothe-angfeoo'nta_ined'oi'Chctwoii’des^.t’h'one:buclwth^foon- 
geft  mhes  one  thing  required,  whiche  is,  that  thebafe^of  the  one,is  equal  to  the 
bale' of  the  other*  No^  th  this  S^propofftio.bdhg  the 

bafeHf.tlie  one  is  equal  to  phebafe  ofth’other.is  thefuppofitionsorthe  thing  ges  ■ 
ne;  which  in  the  former  propofitio  was  the  conclufib.  And  this, that  two  Odes  of 


ft 


Fobs. 


tkeonc  areeqqaH  to  two  fidesof  the  other,  is' in  this  proportion  alfo  afiip^ 
pofkion i, like  as  it  was  m  the  former  propofitionifo  that  it  is  a  thing  geuen  ineis 
ther  proportion.  The  conclufionofthis  proportion  is  that  the  angle  enclo fed 
of  the  two  equal!  fidesof  the  one  triangle  is  equalltothe  angle  enclofed  of  the 
two  equall  fides  of  the  other  triangle:  which  in  the  former  propofiuon  was  one 
of  tbe  things  geuen. 

Philo  and  hisfcholas  demonftrate  this  proportion  without  thehclpe  of  the 
former  propefition,in  this  maner. 


•■'4 

Suppofe  that  there  be  tw©  triangles  A.B.  C  and  D  E  F,  hauing  two  fydes  of  the  one 
equall  to  two  Cycles  of  the  other7namely,v4  B  and  A  C  equall  to  D  E  and  D  F  ,  the  one 
to  the  other,  &  the  bafe  B  C equal  to  the  bafe  E  F .  And  for  that  the  bafe  2?  Cis  equall 
to  the  bafe  E  F,therfore  the  one  being  applied  to  the  Other  they  agree  Place  the  two 
triagles  ABCScDEFi  n  one  &  the  felf  fame  plaine  fuperficies.,&  apply  the  bafe  of  the 
one  to  the  bafe  of  the  other  :But  yet  fo  that  the  triagle  A  B  C  be  fet  one  the  other  fide  of 
the  right  line  £  F,that  the  top  of  the  one  may  be  oppofite  to  the  top  of  the  other .  And 
in  {lead  of  the  triangle  AS  C  put  the 
triangle  E  F  G  as  in  the  figure.  And  let 
D  E  be  equall  to  E  <7, and  F)  F  to  F  G. 

Nowe  then  by  this  meanes  (hall  hap¬ 
pen  diners  cafes  ♦  For  the  line  F  G  may 
fall  diredtly  vpo  the  line  D  F,orit  may 
fo  fall  that  it  may  make  with  the  line 
15  F  an  angle  within  the  figurejor  with 
out.  ,  . 

'  j  /  *  .  1  -  =1  ? 

Firft  let  it  fall  dire&lye .  And  fqraf- 
mqche  as  the  line  D  E  is  equall  to  the 
line  E  6’,anclD  FG,  is  one rightc line: 
therfore  DEG  is  an  Ifofcelcs  triagle: 
and  fo,by  the  fifth  pt-opofition,the an¬ 
gle  at  the  point  D  is  equal  to  the  angle 
at  the  poynt  G :  which  was  required  to 
beproued. 


But  ifit  fall  not  dire&ly ,  but  make  with  the  lineD 
F  an  angle  within  the  figure,  drawe  a  line  from  D  to  G. 
Now  forafmuch  as  E  D  and  E  G  are  equall ,and  the  line 
D  G  is  the  bafe’therfore  by  the  fifth  propofitio,  the  an¬ 
gle  E  D  G  is  equall  to  the  angle  E  G  D  .  Agayrte  foraf¬ 
much  as  D  F  is  equall  to  F  G,and;D  G  is  the  bafe :  ther- 
fore  by  the  fame,the  angle  F  D  G  is  equal  to  the  F  GD: 
and  it  was  proued  that  the  angle  E  D  G  is  equall  to  the 
angle  E  G  Diwherfore  the  whole  angle  ED  F  is  equal  to 
the  whole  angle  FGB  :  whione was  required  to  bepio- 
ueeb 


yin  other  detitt* 
firation  itutend 
ted bj  Philo. 


But  if  the  line  FG  make  with  the  line  D  F  an  angle 
without  the  figure.-draw  a  right  line  without  the  figure 

V.ij.  from 


After  this  de~ 
monftrsstio  thre 
safes  in  this  fr& 
fitters* 


Thefirft  cafiti 


T he fieetsd  cafiti 


The  third  cafi* 


from  thepoyirt  D  to  the  poynt  G.  And  forafmuch  as  D  E  and  E  G  are  equall,  and  DG 
is  the  bafe,therfore  by  the  fifth  propofitioiuhe  angles  E  D  G  and  D  G  E  are  cquall.A- 


-gaine  forafmuch  as  D  F  is  equall  to  F  G,and  D  G  is  the  bale,  therfore,by  the  fame, the 
angle  F  D  G  is  equal!  to  the  angle  F  G  D  .  And  it  was  proued  that  the  whole  angles  E 
3>G  &  D  G  E  are  equall  theonc  to  the  other:  wherfore  the  angles  remayning  E  D  F  & 
EG  F  are  equall  the  one  to  the  other,which  was  required  to  be  proued. 


i  5 


ZmBrn&im* 


DetMOu&ration. 


c>  "  J  ‘  '  -  -•  -  -  i  i  1  ‘  l'.* 

^&The  ^.fP  roblcme.  T  he  p.Propojitton, 

sot  EuA ni  ?:;■?, '  :  nsttt 

To  deulde  a  rcBiline  angle geuenfinto  trn  equal l partes, 

v- t  ■  ■  ‘(y  -  .  ...  ri  ,  . *. .  .  . . ■ , 

V ppofe  that  the  reftiUne  angle  geuen  he  B  A  C  It  is  re* 
qutredto  deuide  the  angle  IB  A  C  into  two  equal  partes. 
In  the  line  A  B  take  a  point  at  ad  aduentures,  let  the 
fame  he  D.And(by  the  third  proportion) from  the  lyne 
1  A  C  cutte  of  the  line  A  E  equall  to  A  ID,  And  (by  the 
firft  petition)  drast  aright  line  from  thepoint  fDto  the 

_ _ _ !  point  E.Andfhy  the  fir  ft  propofitm)vpon  the  line  ID  S' 

defcribe  an  equilater  triangle  and  let  the  fame  be  DFEy  and(by  the  fir  ft  peti • 
cion)  draSoe  a  right  line  from  the  poynte  A  to 
the  point  F.  Then  l fay  that  the  angle  BAG  is  by 
y  line  AF  deuided  into  two  equal  partes. For  for* 
afmuchas  A  V  is  equall  to  A  Ey  and  A  F is  canton, 
to  them  hotktherfore  thefe  noo  D  A  and  AF^are 
equall  to  thefe  two  FA  and  A  Fythe  one  to  the  o* 
ther!But(hy  thefirftpropopfm)the  bafe  D  Fis 
equall  to  the  bafe  EF‘Merfdre(hy  the  ft.propo* 

/ition)the  angle  D  AF  is  equal  to  the  angle  FAE.  & 

Wherfore  the  reBiline  angle  geuen  y  namely  yB 

AC  is  deuide  d  into  tioo  equal  partes  by  the  right  lint  A  F\ Which  ms  requU 
red  to  be  done . 


i .  -  ^  . 


In  this  propofition  is  not  taught  to  deuidc  a  right  lined  angle  into  mo  partes 
then  tw.  albeit  to  demde  anangle/o  it  be  a  right  angle, into  three  partes,  it  it 

L  ‘  >.  not 


ofEmlidesElementes.  FoLip. 

It  is  tmpof 

doc  hard.  And  it  is  taught  of  Vitellio  in  liis  firft  bokc  ofPerfpe£h'ue,tbe  2S,Propo-  fileta  deuide 
fition.Por to  deuide an  acuteangle  into threeequal  partes  ,is(as  faith Proclus)\m-  *2-1CHtere‘,  ■ 
poffible:vnlesttbe  by  the  helpc^orothcr  lines  which  are  ora  mixt  nature, Which  to  three  equail 
thing  Tfjcomedes  did  hy 'fuch  lines  which  are  called  Concoides  linear  ho  fir  ft  ferched  partes  without 
out  the  inuemion,nacurc,&:  properties,  of  fuch  lines.  And  others  did  itby  other  are 

meanes:as  by  the  helpe  of  quadrant  lines  indented  by  Hippies  &  jy/«*w/^«tOther$  0Ja 
by  Helices  or  Spiral  lines  indented  of  Archimedes  .B  ut  thefe  are  things  of  much  dif-  tme. 
ficulty  and  hardnes,and  nothere  to  be  infreated  oft 


Hereagainft  this  propofition  may  ofthe  aduerfary  be  brought  an  -Jfinftance.  *j„-tnfiaKce  ;t 
For  he  may  cauill  that  the  hed  ofthe  equ’i'latertriangle  (hall  not  fall  betwene  the  an  obieftion  or  a 
two  right  lines ,but  in  one  ofthem^or  w  ithout  them, both.  As  for  example.  doubtjvherbj  u 

1  letted  or  trots- 


Suppofe  that  the  angle  to  be  deuided  into  two 
equail  partes  be  B  A  C ,  and  in  the  line  *B  <tA  take  A 

the  poynt  D  , and  vnto  the  line  ©  A  put  the  line  l 

A  E  equalfby  the  third  propofition . )  And  draw  a 
line  fro  D  to  E,  And  vpon  the  line  DE  defcribe(by 
the  firft)  an  equilater  triangle, which  let  be  DEE. 

Now  then  if  it  be  poffible  that  the  point  F  do  not 
fal  betwene  thelines  ABScA  C,then it ihal fal  e- 
ther  in  the  \it\eAB  or  AC,  or  without  them  both. 

Suppofe  that  the  point  F  be  fall  vpon  line  AB,  fo 
that  let  D  F  E  be  an  equilater  triangle.?  Wherfore 
the  line  D  F  is  equal  to  the  line  F  E:  &  the  angles 
at  the  bafe  are  equail,  namely,  the  angles  EDF 
and  D  E  F.  Wherefore  the  whole  angle  D  E  (7  is 
greater  then  the  angle  EDF.  Againe  foraftnuch  as  zA  D  is  equal!  to  A  £,  therefore 
AD  E  is  an  Ifofceles  triangle.  Wherefore  ( by  the  fifth  propofition  )  the  angles  vn¬ 
der  the  bafe  are  equall.Wherfore  the  angle  D  E  Cis  equail  to  the  angle  ED  B  .  But  it 
was  alfo  greatertwhich  is  impoffible.  Wherfore  the  top  ofthe  equilater  triangle  canot 
be  in  the  right  line  A  2?.  And  in  like  fort  alfo  may  we  proue  that  it  canot  be  in  the  right 
Vine  A  C.Wherforefuppofethat  it  be  without  them  both, if  it  be  poffible.  Andforaf* 
much  as  D  F  is  equal  to  F  E,the  angles  at  the  bafe  are  equal,namely,the  angles  DEF  & 
E  D  F. Wherfore  the  angle  D  E  F  is  greater  then  the  angle  E'DV -Wherfore  the  angle  D 
EC  is  much  greater  then  the  angle  E  D  F.But  it  is  alfo  equal  vnto  it.  For  they  are  angles 
vnder  the  bafe  DE  of  the  Ifofceles  triangle  AD  E.  Which  is- impoffible .  Wherfore  the 
poynt  F  (hall  not  fall  without  the  two  right  lines  on  that  fide  .  And  in  like  forte  may  we 
proue  that  it  (hall  not  fall  without  them  on  the  other  fide .  Wherfore  it  fhali  of  neceffi- 
ty  fall  betwene  them :  which  was  required  to  be  proued* 


hied  the  con- 
fins  ft/on ,  or  de¬ 
mon  ftrat  son, 
contajnsth  an 
9 nruth ,  and  an 
impoffibihtj: 
and  ther  fore  it 
mvft  of nece fifty 
be  anfrered  \>n~ 
to, and  the  fill fe- 
hode  thereof 
made  manifefi. 


Theremayalfo  in  this  propofition  bedmers  cafes. (fit  fo  happen  that  there  Diners  cafes  in 
be  no  fpace  vnder  the  bafe  D  E  to  deferibe  an  equilater  triangle, but  that  of  necef-  this  propofition, 
fitie;youmuft  deferibe  it  on  the  fame  fide  that  the  lines  AB  and  A  C  are.  For 
thenthefides  ofthe  equftater  triangle  either  exa&ly agree with  the  lines  AD 
and  A  H,  if  the  faid  lines  A'D  and  AE  beequall  with  the  bafe  D  E.  Or  they  fail 
without  thera'jifthe lines  A D  and A E  beleffe  then  the  bafe  DE-  Or  they  fall 
within  them, if  the  faid  lines  be  greater  then  the  bafe  D  E. 

>  ’  y  r 

-s:n:  o-  rmn  ■  "  a  ::r  :  •-  , 

Firft  let  them  exactly  agree,  And  let©  A  E  be  an  equilater  triangle.  And  in  the  fide  The  firft  cafe, 
A  D  take  the  poynt  G ,  And  from  the  fide  A  E  cut  of  a  line  equal  to  the  line  A  G(  by  the 
third  propofition  Jwhich  let  be  A  fi, And  draw  thefe  right  lines  (j  E,HD  and  GH, and 
AF.Now  forafmuch  as  AD  is  equal  to  AEfzndA  G  vnto  A  /ifttherfore  thefe  two  lines 

F-iift  DAmd 


sjn he  peon  d  cafe. 


TbejirB'Boofy 


A  and  A  Hare  equall  to  thefe  two  lines  £.4 
and  A  Grand  they  contaync  one  and  thefelfe 
fame  angle,  Wherfore  by  the  fourth  propofi- 
tion,the  angle  G  D  H  is  equal  to  the  angle  H  E 
G.  And  the  bafe  ‘D  H  is  equall  to  the  bafe  E  G. 
But  the  line  D  G  is  equal  to  the  line  E  H:wher 
fore  againe  by  the  fourth  propofition ,  the  an¬ 
gle  EGHr  equall  to  the  angle  2)  HG.  Wher¬ 
fore  by  the  fixt  propofition,  thebafeGFise- 
qual  to  the  bafe  H  F  And  forafinuch  as  A  His 
equall  to  A  G,and  A  F  is  connnonjto  the  both, 
and  the  bafe  G  F  is  equall  to  the  bafe  H  F,  ther 
fore  the angleGAF  is  equall  to  the  angle  HA 
F. Wherefore  the  angle G  AH  is  deuided  into 
two  equal!  partes :  which  was  required  to  be 
done. 


B  ut  if  the  fides  of  the  equilater  triangle  fall  without  the  right  lines  BA&A  C,as  do 
the  lines  F  &  £  F,the  draw  a  line  from  F  to,  A  &  produce  the  line  F  A  to  the  point  G. 
Now  forafinuch  as  the  lines  D  F  and  F  E  are  equal. 

Sc  the  line  F  A  is  common  to  them  both,  &  the  ba- 
fes  D  A  and  A  E  are  equall:  therfore(by  the  eight) 
the  angle  D  F  A  is  equal  to  the  angle  E  F  A.  Agatne 
forafinuch  as  D  F  and  F  E  are  equall ,  and  F  G.  is 
common  to  them  both,  and  they  containc  equall 
angles  (as  it  hath  bene  proued )  therefore  (by  the 
fourth)  the  bafe  D  Gis  equall  to  the  bafeGE.  And 
forafinuch  as  A  D  is  equall  to  A  E,and  A  G  is  com-  ; 
mon  to  them  both,  Therfore(by  the  eight)the  an¬ 
gle  PAG  is  equall  to  the  angle  E  A  G  .wherefore 
the  angle  D  A  E  is  deuided  into  two  equall  partes :  0 
Which  was  required  to  be  done. 


•S the  third  cap. 


T<4  dcaide  a. 
re&tiine  angle 
■onto  two  equall 
farts  Mechani- 
Jtatlj. 


Butif  the  fides  of  the  equilater  triangle  fal  with¬ 
in  the  right  lines  B  A  and  A  C,  as  do  the  lines  D  F 
and  F  E,then  againe  draw  aline  from  A  to  F.  And 
forafinuch  as  D  A  is  eq  uall  to  A  E,and  A  Fis  com¬ 
mon  to  them  both,and  the  bafe  D  F  is  equal  to  the 
bafe  F  E: ther fore  the  angle  D  A  F  isf  by  the  eight) 
equall  to  the  angle  E  A  F,  Wherefore  the  angle  at 
the  point  Ais  deuided  into  two  equall  partes,how 
foeucr  the  equilater  triangle  be  placed:  which  was 
required  to  be  done. 


Tbi  s  is  to  be  noted/that  if  a  man  will  me¬ 
chanically  or  readily,  not  regatdyng  demon'-, 
ft  rati  on,  denude  the  forefaid  reftilincangle  B 
AC  ,and  Co  any  other  re<fhlirreanglegeuenwhatfoeii£f, 
into  two  equall  partes3he  {hall  needeonely  with  one  o£ 
peaing  of  the  compafle  tafcen  at  all  adtientures  to  markc 
the  two  pointes  D  andE,  which  cult  of  equal  partes  of  the 
lines  A  B  and  A  C,howfoeuerthey  happen,  and  fo  ma¬ 
king  the  centres  the  two  points  Dand  E,  to  deferibe  two 
circles  according  to  the  openyng  of  the  compafle  .*  and 
fcom  the  point  A  to.  their  interfe&ion,  which  let  be  the 
point  F  to  draw  a  rightline:  which  fliall  deuide  the  an¬ 
gle  B  A  C  into  two  equall  partes.  Add  here  note, 'that 
youfhall  not  nede  to  draw  the  circles  all  whole^but  one* 


of Euclides  Elelnentes . 


Folio. 

ly  a  portion  where  they  cut  thbbiib  the  o'tftttY  As  m  thehgote  here  in  the  end  of 

the  other  fide  put. 


&!  H  £  Vr  V., 


S-. 


...  ;  .  i  n  r  „  ••  •  ^  (  :  ^  ^ 

The  sfProhleme.  f  The  iQ.Tropofition u 


,  VC 


3  von-. 


-  :iU 


T o  deuide  a  right  line  getten  being  finite  (into  tm  equall 

^  !  "vai  31 

partes. 


V/.i  J. 

nt 

.3UO  510; 


Vppofe  that  the  right  linegeuenbe  A  <B  .It  is  required  to  deuide  the 
line  A  B  into  fWo  equal partes.Defcribe( by  the firjl propo/ition)vp * 
on  the  line  AS  an  equila  ter  triangle  and  let  the  fame  he  A  BC.  And 
^  (by  the  former  propofitiohfdeuide  the  angle  AC  B  into  tyi>o  equal l 
partes  by  the  right  line  CD.TbenI fay  that  the  i  > 

right  line  A  B  isdeuided  into  two  equall partes  in 
the  poynt  D.For forafnuch  as  (  by  thefrfipropo* 
ftion)A  C  is  equall  to  CB ,  and  C  D  is  common  to 
the  botktherfore  thefe  two  lines  AC  <&  CD  at e  e* 
qua  l  to  thefe  two  lines  BC<jr  £D(y  one  toy  other, 
and  the  angle  AC Dis  equaUto the  angle  BCD.  t 
VVbetf&re(by  the  ^propofition  )  tfy  bafe  AD  is  :  /  • 
equqll  to  the  bafeBDAVhere fore  the  righte  line  JC~'  p  3 

geuen  A  B,is  deuided  into  VWoe  quail  partes  in  the 
poynt  Dibich  'Was  required  to  be  done 4  .  '  v 

Apollonius' 'teacheth  to  deuide  a  rightlinebeihgfinite  into  two  eq«aih|)4[|tes 
after  this  manner,  1. 

c  r  „u  ....  ,  .  :  vjdi*  0*  ran  *1  /l  J  a 

Suppoie  that  the  right  line  being  finite  ,,  ^  , 

be  AB: which  it  ii  required  to  deUirle  ih?Ji  ■ 
to  two  equal  parts .  now  the  makingthCv .cr; ;•  A  .  vt y 
centre  the  point  A  &  the  fpaoe  A  B  de- 5  t  / 

Icnbe  a  circle.  Again  making  the  centre  ,  .. . 

the  poynt  B  &  the  {pace B  A  defcribeah  'V“ 1  '  ’A”' 

orthetdrclsiand  froxnths rfc,  5  ' 


f£^thitthenl^ht  find  ^ 

othe^fbrthat  they  frd 

the  centres  to  the  circumferences  of  equall  circles.And  forafmuch  as  the  lines  C^AM^ 
A ‘Dare  equall  to  the  lines  C  B  and  B  D.and  the  line  CP  is  common  to  either  of  them: 


-V-  mis 

■••i-  \  oUva-  '  , 

l  .  * 

v  "d  1  -tvi  a'-. 


C  on  ft  ruction* 


Demon^ratioji, 


,nsVtWn\:..i 


■ 

An  other  way  ft 
deuide  a  right 
line  being fnit<t, 
inuented  bj  A* 
pellontus. 


By  this  way  ofdeuidinga  rightline,  into  two 
equall  parts  inuented  by  Apollonius,  it  is  manifcft, 
that  if  a  man  wil  mechanically  ,or  redeIy,not  confi- 
dering  the  demonftrati5,deuicje  the  faid  rightline, 
andfo  any  Tight  line  geuen  whatfoeuer, into  two  e- 
quall:  partes  he nedeonely  to  marke  the  poyntsof 
the  rnterfe&ions  ofthecirc'les,  8C  to  draw  adinefrS 
the  fa/d  interfe&ions, which  fhall  deuide  the  right 
line  gcuen  into  two  equall  partes  :  as  in  the  figure 


X 


X 


5^  The  6.KProblemc.  The  IvTropoJitim. 


Vpm  a  right  tme geuen,  torayfe  tyfrom  a  poynt geuen  in  the 
Jameltne  a  perpendicular  line. 


:  -  Vi  j 


V Dpofe  that  the  right  line  geuen  he  A  '!*>><&  let  the  point 
in  it  geuen  be  Clt  is  required ftotn  the  poynte  C  to  rayfe 
Vp  Vnto  the  right  line- A  a  perpendicular  line  .  Take 
in  the  line  A  C  a  poynt  at  all  aduentures3  &  let  the  fame 
beD>and(bythe$ipr.6pojition  )  put  VntoDC an  equall 
j  line  C  E.And  by  the firftpropofttionftpon  the  line  D  E 
fteferibe  an  equilater  triangle  FID  E*  ^dra'W  a  line  fro 


FtoC4  Then  I  fay  that  Vnto  the  right  Untgeuen 
A  'Bgmdfrom  the  poynte  in  itgeuen ,  namely jC  is 
ray/edvp  a  perpendicular  line  FC.  For fora/much 
as  DChequalto  CE the  lineC  F  iscom&n  to 
them  botkeherfore  thefe  ttto  DC  and  C  F^are 
qua l  to  thefe  nto  EC  ejr  C Ffthe  one  to  the  other : 
and(hy  the  fir  ft  propofition)the  bafeD  F  is  equal 
to  the  bafe  E  F :  wherefore  (by  they  tpropo  fit  ion) 
the  angle  DCF  is  equaUto  the  angle  ECF:  and  ^ 
they  be  fide  angles&ut  lphe  a  right  line  ftanding 
ypona  right  line  doth  make  the  two  fide  angles  equall the  one  to  the  other ^ether 
of tboje  equall  angles  if  {by  the.\o,definition)a  right  angleg^r  the  line  ftanding 
Tpon  the  right  line  is  called  a  perpedicular  Ime.VVherfore  the  angle  D  C 
wangle  F  C  E  are  right  angles.VVherfore  Vnto  the  right  line geue  A  fro 

Cjtxayifedjrp  ttotCffcp.bfcb  "teas  required  to  be 

done  4  '■  ■ ' 

:m3iihlppr  h  -  no > nmoa zi U 3 &oH silt inu..C  . ■,  ;Unpv.-ifiG?K 

Although  thepoynte,  geuen  flipuld  be  fet  in  one  of  the  endes  of  the  rightc 
fihe 'jgeufci)»it  is'  eafy  fp  tfdt  it^s  it,  Was-heforc  ;  For  producing  the  line  in  length 
iiqpithe  poynt  by  tfie  fecorid  peticuphjy  pfirnay  wprkeas  you  did  before.  But  if 
one  require  to  ereft  ^  r%^t  line  pcrpendicjqlarly  from  thepoy  nt  at  the  end  of  tfi e 

■  ;  iotfisiifipet  tr  rbidwia. ai gsjiiq Ifcu  i|*“* 


ofEnclides  Elementes. 


FoUu 


Ivne,  without  producing  the  rightlyne,  thatalfo  may  well  bee  done  after  thy* 
raaner,  '  ■  ■  •  -  l  i  ■ 


Suppofe  that  the  right  line  geuen  be  A 2?,&  let 
the  point  in  it  geuen  be  in  one  of  the  endes  theroB 
namely  ,in  A  .And  take  in  the  line  A  B  apoint  at  all 
aduenturcs,andletthefamebe  C.  And  from  the 
faidpoint  raifevp(by  the  fore  Paid  propdution)vn- 
toABa  perpendiculer  Iine,which  let  be  CEt  And 
(by  the  3  .propofition)from  the  line  CE  cut  of  the 
line  C  D  equal!  to  the  line  C  A,  And  (  bythe  9  .Pro- 
pofitionjdeuide  the  angle  AC  D  into  two  equall 
partes  by  the  line  CD,  And  from  the  point  D  raife 
vp  vnto  the  line  CE  a  perpediculerline,D  F,which 
let  concurre  with  the  line  C  Fin  the  point  F,  And 
drawe  a  right  line  from  F  to  A ♦  Then  I  fay  that  the 
angle  at  the  pojnta^  is  a  right  angle.  For,  foras¬ 
much  as  D  Cis  equall  to  Ce^f,  and  CF  is  common  to  them  both3and  they  containe  e- 
quall  angle§(for  the  angle  at  thepoint  Cis  deuided  into  two  equall  partes )  therefore 
fby  the  4.  Propofition)  the  line  D  F  is  equall  to  the  line  F  A,  and  fo  the  angle  at  the 
point  A  is  equal  to  the  angle  at  the  pointD,  But  the  angle  at  the  point  D  is  a  right  an¬ 
gle.  Wherfore  alfo  the  angle  at  the  point  zsfisa  right  angle.Wherefore  from  the  point 
A  vnto  theline  raifed  vp  a  perpendiculer  line  ^F,withoiitproducingthe  line  A 

B .  Which  was  required  to  be  done. 


F 

D 

\ 

L 

3 


An  other  cafe  m 
this  proportion* 

Ctnsimffkn,  ; 


Demonftration, 


Appollonius  teacheth  to  ray  fe  vp  vnto  a  line  geuenjfrom  a  point  in  it  geuen, 
a  perpendiculer  line^fter  this  maner*  : 

•  v  *.  .W--C  D  '  •  •  : 

Suppofe  that  the  right 
line  geuen  be  A  B.  And 
let  the  point  initgeue, 
be  C,  And  in  theline  a A 
C,take  a  point  at  all  ad- 
uetures,&kt  the  fame 
beD.  Andfrothelyne 
CSjCMtofaline  equall 
to  the  line  CD,  whiche 
let  be  C £.-  and  makvng 
the  centre  D, and  the 
{pace  D  E,  deferibe  a 
circle.  Andagaine  ma¬ 
king  the  centre  C,&  the 
fpace  E  P ,  deferibe  an 
other  circle,  and  let  the  point  of  their  interfedion  be  F,  And  draw  a  right  line  from  F  to 
C  .Then.  I  fay  that  the  line  F  C  is  creded  perpendiculerly  vnto  the  line  o 4  B.  For  drawe 
thefe  lines  F  D  and  FF-.  which  flial  by  the  definition  of  a  circle  be  either  of  them  equal 
to  theline  D  E.  and  thcrforc  (by  the  firfi:  common  fentcncc)  are  equall  the  one  to  "the 


- - 

y  > 

A 

i  / 

f 

v  ■ 

\A  P\  -  C 

Another  Way 
to  ereli  a perpf- 
dictriar  Itnc  its* 
vented  by  Ap» 
palon'mt. 
Ctnftruttknp 


■  -c . 


D  wjafefj'ssJ 


wherfore  they  are  right  angles,  Wherfore  the  line  CF  is  eroded  perpendiculerly  vnto 
theli^^£fromthepointC(4.whichwasrequiredtobedone- 


-  !  v 


By  tltis  wav  oferedii?ga  perpendi^uletiineinuented  by  Appollonius,  it  is 
ilfpmanifc'ft,fhai;  if  a  man  y  ill  mechamcaliy.withoutdemonftration  xreci  vnto 

‘  -  *  . t  '  G-h  ‘  Mk 


Hovtto  trtff  a 

ferpendietilatt 
ttne  meehmi* 


Cexfir&ftian. 


Ztcmonjlration. 


TbefirH'Boofy 

a  line  geuen  from  a  point  gcue  in  ita  perpendi- 
culeriine:  he  needeonely  on  either  fide  of  the 
pointe  geuen, to  cut  of  equall lines  :  andfo  ma¬ 
king  either  of  the  cndes  ofthe  faid  lines  (either 
ofth’endes  I  fay  , which  haue  not  one  point  cos 
mon  tothemboth)  the  centres,  and  thefpace 
both  the  lynes  added  together,  or  wider  then 
both,or  at  the  left  wider  the  one  ofthem,to  des 
fenbethofe  portions  ofthe  circles  wherethey 
cutthe  one  the  other,  andfromthepoint  ofthe 
interfc&ion  to  the  point  geuen^o  drawalync, 
which  {hall  be  perpendicular  rntothe  lync  ges 
tie:  as  in  the  figure  here  put  it  is  manifeft  to  fee« 

T'heyfProbleme .  7 he  iiSPropofition. 

V Mo  a  right  line  geuen  being  infinite ,  andfroM  a  point geuen 
not  being  in  the  fame  line  Jo  draw  a  perpendicular  line • 

Et  the  right  line  geuen  he* 
ing  infinite  he  JBjjr  let y 
point  geuen  not  being  in  the 
faid  line  A  (B,be  Ct  It  is  re* 
quiredfrom  the  point  gene, 
namely, C Jo  draVe  Vnto  the 
fright  line  geuen  A  B,a  per * 
pendiculerline.Take  on  the  a 
other fyde  ofthe  line  jfB  ( namely ,  on  that  fide 
therein  is  not  the  pointe  C)  a  pointe  at  alladuen* 

tures,and  let  the  fame  be  S).  Und  making  the  centre  C,and  the /pace  C  de* 
fcribe( by  the  third peticion)a  circle, and  let  the fame  heEF  Gjohich  let  cutte 
the  line  A  B  in  the  pointes  E  and  G.And  (by  the  x.propofition )  deuide  the  lyne 
E  G  into  two  equal  partes  in  the  point  ELAnd(by  the  fir/l  peticion)draTV  thefe 
right  lines, C  G,C  H,and  C  E.Thenl fay, that  Vnto  the  right  line  geuen  A  Bjsr 
from  the  point  geuen  not  being  in  it, namely,  C,  is  dralven  a  perpendiculer  lyne 
C  H-Forforafinuch  as  G  His  equall  to  HEy^dAiCis  commonto  them  both: 
the r fore  thefe  two  fydes  G  Hand  HC,are  equall  to  thefe  Ctoo  fydes  EH&H 
£  the  one  to  the  other:  and  (by  the  if  definitio)the  bafeCG  is  equal  to  the  baft 
C Either fore  (by  the  $.propofition)the  angleCHGis  equall  to  the  angle  C 
HE:  and  they  are  fide  angles i  butis/hen  aright  line  (landing  Vpon  aright  line 
maketh  the  tTVofyde  angles  equall  the  one  to  the  other, either  of  thoft  equall  an* 
Aes  is(  by  the  \ot definition )  a  right  angle ,  and  the  line  (landing  Vpon  the  fay  de 
right  hne  is  called  a  perpendiculer  Une.Wherfore  Vnto  the  right line  gene  A(B , 
and  from  the  point  geuen  Cftobicb  is  not  in  the  line  A  BfisdraTMi  a  perpendicu* 
let  fine  C  H:  T&hich  as  required  to  be  done ,  Thiy 


>c 


x 


of  Enclitics  Elementes. 


Fol.zz. 


Thi  s  probleme  did  Oenopides  firft  fictde  out,confidering  the  neceflfary  yfe  thers 
of  to  the  ftudy  of  Aflronomy  ♦ 

There  are  two  kindes  ofperpendiculer  lines:  wherofonc  is  a  plaine  perpen* 
diculer  lynCjthe  other  is  a  folide*  A  plaine  pcrpendiculer  line  is  , when  the  point 
from  whence  the  perpendiculer  line  is  drawer),  is  in  the  fame  plaine  fuperficies 
with  the  line  wherunto  it  is  a  perpendicularv  A  folide  perpendiculer  line  is,  whe 
the  point, from  whence  the  perpendiculer  is  drawne,is  onhigh,and  without  the 
plaine  fupcrficics.So  that  a  plaine  perpendiculer  line  is  drawento  a  right  line:  8c 
afolide  perpendiculer  line  is  drawn  to  a  fuperficies*  A  plaine  perpendiculer  line 
caufeth  right  angles  with  one  oriely  line,  namely,  with  that  vpon  whome  it  fal- 
leth.But  a  folide  perpendiculer  line  caufeth  right  angles, not  only  with  one  line, 
but  with  as  many  lynes  as  may  be  drawn  in  that  fuperficies, by  the  touch  cherof* 
This  propofition  teachcth  to  draw  a  plaine  perpendiculer  line,  for  it  is  drawn  to 
one  line,  andfuppofed  to  bc  in  the  fclfe  fame  plaine  fuperficies. 


f  There  may  be  in  this  propofition  an 
other  cafe.For  if  it  be  fo,  that  on  the  o- 
ther  fide  of  the  line  cA  B ,  there  be  no 
fpace  to  take  a  pointe  in  butonely  on 
that  fide  wherein  is  the  point  C .  Then 
take  fome  certaine  point  in  the  line  s A 
2?,which  let  be  D.And  making  the  cen¬ 
tre  the  pointC,  and  the  fpace  CD,de- 
feribeapart  of  the  circumference  of  a 
circle, which  let  beD  E  F.-which  let  cut 

the  line  A  Bin  the  two  pointesD  and  _ 

F.And  deuide  the  line  D  F  into  twoe-  A 
quail  partes  in  the  poyntH.  And  draw 
thefe  lines  C  D,  C  H  and  C  F,  And  for- 
afmuch  as  D  H  is  equal  to  H  F,and 
C  H  is  common  to  them  both,  and 
CD  is equall toCFfby  the  iy.de- 
finition:)therfore  the  angles  at  the 
point  H  are  equal  the  one  to  the  o- 
ther(by  the  8, propofition:)  &they 
are  fide  angles,  wherefore  they  are 
right  angles  .Wherfore  the  line  C 

H  is  a  perpendiculer  to  theline  D  _ _ 

F.  But  if  it  happen  fo  that  the  circle  A 
which  is  deferibed  do  not  cutte  the 


lyne,  but  touche  it,  then  takyng  a 
point  without  the  point  E,name- 

ly,thc  point  G,and  making  the  centre  the  poiht  C,and  the  fpace  CG,dcfcribe  a  part  of 
the  circumference  of  a  circle:  which  fhall  of  neceflkie  cut  the  line  AB:  andfomay  you 

proceede  as  you  did  before.  As  you  fee  in  the  fecond  figure. 


$■&?  The  6 . 1‘heoreme.  The  1 3.  Tropofotion - 

W  hen  a  right  line  founding  ypon  a  right  line  ma^eth  any  an - 
gles:  thofe  ang les  jhall  be  either  two  right  angle  s3or  equall  to 
two  t  ight  angles* 

Supfofi 


Oenopides  tha 

firjl  in u  enter  of 
this  probleme. 

Two  ktndes  of 
perpendiculer 
lines, namely  y<t 
plaine  perpends ■» 
culer  line  andst 
folide .  ■' 


This  propoftson 
teacheth  to 
draw  a playne 
perpendiculer 
line. 

*in  other  cafe  m 
this  propofition » 


ConftuHim^ 


Demotfrntsossh 


.\SV,v. : 

AA» 


<FbeJir$cBoo%e 

Vppofe  that  the  right  line  A  B flanging  Vppott  the  right 
line  CD  do  make  thefe  angles  C  BA  and  ABD.  Then 
l fay,  that  the  angles  CBA  and  AB  D  are  eyther  two 
right  angles, or  els  e quail  to  two  right  angles  Jf  the  angle 


(by  the  n  propoption )Vnto  the  right  line  CD, and  front 
the  pomtegeuen  m  it,  namely,  B,a  perpendiculer  line  B  \E. W her  fore n 


Conpruclio#^  .  . 

Demonstration 

x. definition)  the  angle  C  B  Eand  E  BD  are  right  angles , ISloTfr  forafmuch  as 
the  angle  C  B  Efts  equall  to  thefe  Etoo  angles  CBA  and  ABE,  put  the  angle 
EBD  common  to  them  both\'it>her fore  the  angles  CBE  and  E  B  D,are  equal 
to  thefe  three  angles  C  B  A  A  B  E,and  E  B  D.Agayne  forafmuch  as  the  angle 
DB  A  is  equall  vnto  thefe  two  angles  D  B  E  and  E  B  A,put  the  angle  A  BC 
common  to  them  both^herfore  the  angles  DBA 
and  A  BC,are  equal  to  thefe  three  angles, DBE, 

E B  A, and  ABC.  Audit  is proued that  the  an* 
gles  CBE  and  EBD  are  equal  to  the  felfefame 
three  angles:  but  thinges  equall  to  one  &  the  felf 
fame  thing, are  alfo(by  the firU comma  fentence) 
equall  the  one  to  the  otbe.VVberfon  the  angles  C 
B  Eand  E  B  D  are  equall  to  the  angles  DBA  & 

A  B  C,  But  the  angles  C  B  Eand  EBD  are  two 
right  angles^berfore  alfo  the  angles  D  B  A  and 
A  BC  are  equall  to  two  right  angles .  Wherfore'^hen  a  right  line  j landing  Vo 
pon  a  right  line  maketh  any  angles:  thofe  angles  jhalbe  either  ttoo  right  angles , 
or  equall  to  right  angle si^hich  was  required  to  be  demonfirated. 

An  othet  demonftration  after Telitarius. 

Suppofe  that  the  right  line  A  B  do  {land  vpon  the  right  line  CD.  Then  I  fay,that  the 
jn  other  Je-  tnrQ  ang{es  J.  g  C  and  ABD,  are  either  two  right  angles,Or  equal  to  two  right  angles, 

man$ram»afz  Fq„  bfi 

perpedicuJervntoC2):  the  is  it  manifeft,  that  they  are  right  an  glesfby  the 
ter PeiturtfM.  conuerfion  0f thc  definition JButifit  incline  towardesthe  end  C,thenfby  the  i i.pro- 

pofitionjfrom  the  point  £,ere<5l  vnto  thelineCDaperpcndiculerline5£,By  whiche 
conftruftion  the  propofitio  is  very  manifeft.For  forafmuch  as  the  angle  A  B  D  is  grea¬ 
ter  then  the  right  angle  D  B  E  by  the  angle  AB  E,  and 
the  other  angle  ABC  is  leffe  then  the.  right  angle  C 
BE  by  the  felre  fame  angle  ABE  :  if  from  the  greater 
bee  taken  away  the  excelfe,  and  the  fame  bee  added  to 
the  leffe,they  Hull  be  made  two  right  angles.  That  is,if 
from  the  obtufe  angle  ABD  be  taken  away  the  angle 
tAB  E,  there  Ihal  remaynethe  rightangle‘X>££,  And 
then  if  the  fame  angle  ABE  be  added  to  the^cute  an¬ 
gle  CB*A,  there  {hail  bee  made  the  right  angle  CB  E. 

Wherefore  it  is  manifeft,that  the  two  angles,  namely, 
the  obtufe  angle  oAB  D,8c  the  acute  angle  <>AB  C, are 
equall  to  the  two  right  angles  CB  E  and  DBE'  which 
was  required  to  be  proued. 


. 

>1 

/ 

/ 

I 

C 

FoLip 

The  7.  Theoreme.  The  14..  Tropojition • 

fffvnto  a  right  line*  and  to  a point  in  the fame  linefbe  dram 
Wo  right  lines,  net  both  on  one  and  the  fame  fide, making  the 
fide  angles  equall to  Wo  right  angles  :  thofe  Wo  right  lynes 
\e  directly  one  right  line. 


rs:v  :^-y= 

Kto  the  right  line  A®,&  ^ 

t-o  j>  point  in  it  ®,  let  there 

be  drawn  two  rightlines  ® 

C,and®®,  Vnto  contrary 

.fides,  making  the fy  dean* 

\gles, namely  ,A ®  C&A® 

®}equall  t<r  two  rbbt  an* 

rrh>  r  'VliPti  T  f/Muth/it  ii  ri.O'frt  C  £> 

x> 


lines®®  and  ®C  make  both  one  right  line.  For 
ifC  Band®®  do  not  make  both  one  right  line  Jet 

the  right  line  ®  E  befo  drawn  to  ®  Cjhat  they  both  make  one  right  linegfoW 
forafmuch  as  the  right  line  A  ®  / Undeth  Vpon  the  right  line  C  ®  Esther  fore  the 
angles  A®  C  and  A®  E  are  equall  to  two  right  angles  (by  the  1 $  propofition) 
®ut  (by  fuppofition )  the  angles  A  ®C  and  A®®  are  equall  to  two  right  an? 
gks:Wherfore  the  angles  OBJ, and  A  ®  E,are  equall  to  the  angles  C®4,and 
A®®',  takeaway  the  angle  A®  C, which  is  common  to  them  both ,W her fore 
the  angle  remay  ning  A®  EJs  equall  to  the  angle  remaining  A®  ®y  namely , 
the  lefie  to  the greater  which  is  impofiiblc. Where  fore  the  line  ®E  is  not  fo  di * 
reftly  drawen  to  ®  Cy  that  they  both  make  one  right  line.  In  like  forte  may  we 
prone, that  no  other  line ,  be  (ides  ®  ®,  can  fo  be  draWne.VF h  erf  ore  the  lines  C 
®  and  ®  ®,make  both  one  right  line, If therfore  Vnto  a  right  line,<&  to  a  point 
in  the  fame  line, be  drawn  two  right  lines. pot  both  on  one  and  the  fame fidejna  » 
king  the  fide  angles  equall  to  two  right  angles:  thofe  two  lines Jhal  make  dire  ft* 
ty  one  right  line:  which  Was  required  to  be  proued.  . 

An  other  deraonftrati on  after  Pelitariiis. 

Suppofc  that  there  be  a  right  line  A  B,  vnto  whofe  pointe  B,  let  there  be  drawen 
two  right  lines  C  B  and  BD, vnto  contrary  fides : and  let  the  two  angles  C  3  A, and  D  3 
Ay be  either  two  right  angles.,  or  equall  to  two  right  angles  »Then  I  fay  ,  that  the  two 
lines  CB  and  BD ,  do  make  dire&ly  one  right  line, 
namely0CZ>.  Forifthey  do  not,th6  let  be  fo  drawn  . 

vnto  CB,  that  they  both  make  dire&ly  one  right  line  Ai 

CB  E:  which  ihallpaife  either  aboue  the  line  2?  Dy  or 
vnder  it .  Firft  let  it  pafTe  aboue  it.And  for  as  much  as  j 

the  two  angles  CB  A  and  ABE3avc(  by  the  former  pro-  j  -E 

pofition^  equall  to  two  right  angles,  and  are  apart  of 
the  two  .angles,  CB  A  and  A3  X>:but  the  angles  C3A 
and  A  B  D  are  by  (fupp  o  fi  tio  n)  e  q  u  a  1 1  alfo  to  two  right 
angles:  therefore  the  parte  is  equall  to  the  whole  which  ___ 
is  impoffible.  And  the  like  abfurditie  will  follow  if  CB  ^ 

G,iii,  E 


leading  ta  jin 
alfurdtue. 


Another  (ltd 
mor.Firatjanaf* 
ter  Pehtarim.r 


a 


o 


pemtnftratioH. 


'Thales  MilepUi 
the  firf  trtn  en¬ 
ter  of  this  pre¬ 
pays  ion. 

No  ean$ruci  'fon 
in  this  propof- 
lion. 

What  hedan- 
pits  are. 


The  eonnerfeof 
this  prepoftio  of 
ter  Peis  tart  as. 


Thefirjl^oofy 

E  pafle  vnder  the  line  2  D.-namcly^that the  whole  Ihalbe  equall  to  the  parttwhich  is 
alfo  inipoflible.  Wherefore  CD  is  one  right  line:  which  was  required  to  be  proued. 

T he  8.  Theoreme.  The  ijSPropojttion* 

fftwo  right  lines  cut  the  one  the  other:  the  bed  angles Jhal  be 
equal  the  one  to  the  other* 

Vppofe  that  the/e  Wo  right  lints  A  B  and  CD,  do  cat  the  one  the  00 
ther  in  the  point  Et  Then  I  fay  y  that  the  angle  A  E  Cjs  equall  to  the 
angle  IDEE.  Eor  fora/much  as  the  right  line  AEJlandeth  Vpon  the 
right  line  l D  Cy  making  thefe  angles  C  E  A, and  A  E  D:  therefore(hy 
the  1 3 *  propo(itio)the  angles  C  E  A, and  A  E  D,are  equall  to  Wo  right  angles ♦ 
Agayneforafmucb  as  the  right  line  D  E,flandeth 
Vpon  the  right  line  A B,  making  thefe  angles  A 
E  Dyand  D  E  (B:therfore(  by  the  fame  propofti • 
on) the  angles  A  E  D,and  D  E  B,are  equall  to 
Wo  right  angler,  audit  is prouedy  thatthe  angles 
CEA,and  A  ED,  arealjo  equall  to  Wo  right  an* 
gles.  VVherfore  the  angles  CEA,and  A  E  D,are 
equall  to  the  angles  A  ED,and  D  E  BtTake  a • 

Way  the  angle  A  E  D  ffchicb  is  common  to  them 
hath.  Wherefore  the  angle remayning  CEAyis 
equall  to  the  angle  remayning  V  E  B,  And  in  like  fort  may  it  he  proued  ,  that 
the  angles  C  E  ByandDE  A%are  equall  the  one  to  the  other .  If  therefore  Wo 
right  lines  cut  the  one  the  other y  the  hed angles  jhalbe  equall  the  one  to  then* 
ther :  Which  Was  required  to  be  demonstrated. 

Thides  Mtkfws  the  Philosopher  was  the  firft  inucntcrof  this  proportion,  as 
but  yetit  was  firft  demonftratedby  Euclide.  And  in  it  there  is 
no  confl ruction  at  all*  For  the  expoiltion  ofthe  thing  gcue,is  fufficientinough 
for  thedemonftration*  - 

//^^/^areappofite  angles, caufcd  ofthe  interfe&ion-of  two  right  lines: 
and  are  to  called,becaufe  the  heddes  of  the  two  angles  are  ioyned  together  ia 
one  pointe* 

The  conuerfe  of  this  proportion  after  Telitarius . 

Iff  after  right  lines  being  dr  Often  from  one  point,  do  make  fofter  angles,  offthtch  thetfto  oppo- 
fite  angles  are  equall;  the  two  oppofite  lines Jhalbe  dr  often  direttly,  and  make  one  right  line . 

Suppofe  that  there  be  fower  right  lines  A  B,  A  C,  A  D,  and  AE,  drawen  from  the 
poyntA,makingfower  angles  at  the  point  A:  of  which  let  the  angle  BAG  be  equall  to 
the  angle  DAE,  and  the  angle  B  A  D  to  the  angle  CAE.  Then  I  fay,  thatB  E  and  C  D 
are  oneiv  two  right  lines:  that  is^the  two  right  lines  BA  and  A  Bare  drawen  dire&ly, 

and 


0 


mentes • 


FoLi\. 


anddco  make  one  right  line,  andlikewfie  the  two  right 
lines  C  A  and  A  D  are  drawen  diredtly  ,and  do  make  one 
right  line.  For  otherwife  if  it  be  poiTible  ,  let  £  £  be  one 
right  line,  and  iikevvife  let  C  G  be  one  right  line.  And  for- 
almuch  as  the  right  line  £  A  dandeth  vpon  the  right  line 
C  G ,  therefore  the  two  angles  EAC  and  EA<j>  are  (  by 
the  13  proportion )  equall  to  two  right  angles.  And  for- 
afmuch  as  the  right  line  G  A  dandeth  vpon  the  right  line 
£ £:  therefore  (by  the  felfe fame )  the  two  angles  EAG 
&ndF  A  G,zre  alfo  equall  to  two  right  angles.  Wherefore 
taking  away  the  angle  EAG ,  which  is  common  to  them 
both, .the  angle  £  A  Cdhall  (  by  the  thirde  common  fen- 
tence)  be  equall  to  the  angle  £  A  G:  b  ut  the  angle  £  A  C 
is  fuppofed  to  be  equall  to  the  angle  BAD.  Wherefore 
the  angle  BAD  is  equall  to  the  angle  FAG ,  namely  a 
part  to  the  whole:  which  is  impoffible .  And  the  felfe 

fame  abfurditie  will  follow,  on  what  fide  foeuer  the  lines  be  drawen.  Wherefore  B  £  is 
one  line,and  CD  alfo  is  one  line:  which  was  required  to  be  proued. 


Demouffratim 
leading  to  a>» 
abfurditie. 


The  fame  comer fe  after  T  rocks. 


If  vnto  a  right  line ,  tend  to  a  point  thereof  he  drawen  two  right  lines ,  not  on  one  and  the  fame  fide , 
in fitch  fort  that  they  make  the  angles  at  the  toppe  equall:  thofe  right  lines Jhalhe  draWen  dir  Ally  one 
to  the  other ,  and  jhalmake  one  right  line. 

Suppofe  that  there  be  a  right  line  A  B,and  take  a  point  in  in  C.  And  vnto  the  point 

in  it  C,  draw  thefe  two  right  lines  CD  and  CE  vnto  contrary  fides,  making  the  angles 
at  the  hed  equal,  namely,  the  angles  A  CD  and  B  CE.  Then  I  fay,  that  the  fines  CD 
and  C  E  are  drawen  dire&ly,  and  do  make  one  right  fine .  For  forafmuCh  as  the  right 
line  CD  Handing  vpo  the  right  fine  A  B,d  oth  make  the  angles  D  C  A  and  DC  B  equall 
to  two  right  angles  (by  the  1 3  propofition  :)and  the  angle  DC  A 
is  equall  to  the  angle  £  C£:  therefore  the  angles  DCB  and  B  fiE 
are  equal  to  two  rightangles.  And  forafinuch  as  vnto  a  certayne 
right  line  £  C,and  to  a  point  thereof  C,are  drawen  two  right  fines 
not  both  on  one  and  the  fame  fide,  making  the  fide  angles  equall 
to  two  rightangles,  therefore  (by  the  1 4, 'propofition  j  the  lines 
C  D  and  C E  are  drawen  dire&ly,  &  do  make  one  right  line,which 
was  required  to  be  proued. 


The  fame  cbm - 
eterfe  after  Peli*> 
tariusgvhtch  it 
demonflratid 
dirt  Illy, 


The  fame  may  alfo  be  demonftrated  by  an  argument  lea¬ 
ding  to  an  abfurditie.  For  if  CE  be  not  drawen  dirc&ly  to 
C D,  fo  that  they  both  make  one  rightline,  then  (if  it  bee 
poffible)  let  CF  bee  drawne  dire&ly  vnto  it.  So  that  let  D 
CF  be  one  right  line.  And  forafinuch  as  the  two  ri"ht  lines 
AB  and  £>£docutte  the  one  the  other,  they  make^the  hed 
angles  equall  (by  the  1 5.  propofition)  Wherefore  the  an¬ 
gles  A  CD  and  B  C Fare  equall  j  but  (byfuppofition)  the 
angles  A  CD  and  B  C  S  are  alfo  equall .  Wherefore  (by 
the  firft  common  fentencej  the  angle  B  CS  is  equall  to  the 
angle  B  CF:  namely,the  greater  to  the  lefle :  which  is  im- 
polfible.  Wherefore  no  other  rightline  befidtes  C£is  dra¬ 
wen  dire<% to  CD.  Wherefore  the  fines  C D  andCFare 
dram*  di.re£tly,andmake  one  right  fine;  which  was  requi¬ 
red  to  beproued. 


The  fame  ct7t* 
uerfe  after  Pro* 
tlut  demonflra- 
tedindirehlj. 


The  fir fl  ^Boo^e 

Ww*Corrol-  ,  Of  this  fiuetenthPropofition  followeth  a  Corrollary.  Where  note  that  a- 
*??"•  CVc/%  is  a  Proportion,  whofedcmonfkation  dependeth  of  the  demonftration 
of  an  other  Proportion,  and  it  appeareth  fodenly,as  it  were  by  chance  offering 
it  felfe  vnto  vs:  and  therefore  is  reckoned  as  lucre  orgayne.  The  Corollary  which 
followeth  ofthis  propofition,  is  thus* 


A  Corrollary 
fi  Mowing  ofthis 
progefittan. 


If fetter  right  lines  cut  the  one  the  other:  they  make  fitter  angles  equall  to  finer  right  angles. 

This  Corollary  gauc  great  occafion  to  finde  out  that  wonderful  propofition  in* 
uented  of  pithagdras, which  is  thus. 


A  TVonderfull 
propofitien  in- 
■uented  by  Pi- 
thagoras . 


Only  three  kindcs  of figures  of  many  angles ,  namely ,an  eqmlater  triangle ,  a  right  angled  figure, 
ofj  ottcrfid.es>  and  a  figure  of fixefidesjoaumg  e quail fiides  and  equal  angles }  canfill  the  ttholefpate 
■about  a  point gheir  angles  touching  the  fame  point.  .  -■ 


Euery  angle  ofi 
an  equiUter  tri 
angle  is  equal  to 
fwo  third  partes 
of  a  right  angle 


Euery  angle  ofi 
a  fixe  angled  fi¬ 
gure  is  e quail  to 
a  right  angle, 
nnd  to  a  third 
part  of  a  right 
angle. 


xj-  •  •»  v* vovV  \ 
-  v  <  \  N  *.i  V‘>  ^ 

-}.  »  *. V/  *  WiYVU  ♦,  Y  » Y 

.  •  lir,'  s* V* 


Euery  angle  of  an  equilater  triangle  contay- 
neth  two  third  partes  of  a  right  angle  ;  fixe  ty  roes 
two  thirdes  of  a  right  angle  make  fower  nghc  an? 
gles.  Wherefore  fixe  eqmlater  triangles  fill  die 
whole  fpace  about  a  point  which  is  equal  to  fower 
right  angles,as  in  the  i, figure.  Alfo  euery  angle  of 
are&angle  quadrilater  figure  is  a  right  angle:wher 
fore  fower  of  them  fill  the  whole  fpaceas  in  the  2. 
figure*Eue«y  angle  otafixe  angled  figure,  i's.equal  ' 
to  a  right  angle,  and  moreouertoathirdparcofa 
tight  angle.  But  a  right  angle,and  a  third  part  of  a 
right  arigkjtake  thre  times,make  4,right  angles: 
wherefore  three  eqmlater  fixeangled'figufes.fill 
the  whole  fpaee  about  a  point:  which  fpace (by 
this  Corrollary)  is  equal!  to  fower  right  angles:  as 
in  the  third  figure.  Any  other  figure  of  many  fids, 
howfoeueryou  ioyne  the  together  at  the  angles, 
flial  either  want  of  tower  anglesjor  exceede  them . 
By  this  Corrollary  alfo  it  is  manifefi  that  • 
sf  mo  then  two  lines,  that  is,  three*  or 
fower,  or  how  many  feeuer  do  cut  the 
one  the  other  in  one  point,  all  the  an¬ 
gles  by  them  made  at  the  point  fhalbe 
equal! ,  to  fower  right  angles., For  they  r. 
fill  the  place  of  fower  right.angl.es.  And 
it  is  alfo  manyfeft,  that  the  angles  by 
thofe  right  lines  made  are  double  in 
number  to  the  right  lines  which  eutte 
the  one  the  other.  So  that  if  there  be 
two  lines  which  cut  the  one  the  other, 
the  are  there  made  fower  angles  equal! 
to  fower  rightanglcs;  but  if  thre,  theta 


•jt:u 


ar 


there  made  fixe  angles ;  if fowetjeightanglesjand  fp  infinitly,Eor  euer  th« 
mukicu.de,  or  number’  of  ofthcangles  is  dubled  to  the  multitude  of  the  rq*hf 
lines  which  cut  the  one  the  other.  And  as  theaogles  increafe  in  multitude,  £0 

diminifii 


Of  nuciian  jzimemes .  FoL  25. 

dimmififi  they  is  cteuided  is  alwayes  one  and 

thefelfeia'mle'thing;,  natndyyfbyer  rijghc’aiigtes,  ' 

l .  :  ’  .  u  ' 

cfke  9  SI  heoreme.  The  i6SPropofition. 


Whenfoeuer  in  any  triangle,  the  lyne  of  one  fyde  is  drawn 

„  ,  .  «  >  »  r  .  1  /»  11  *  1 


forth  in 
one  of  the  Wo 


reater 


vni- 


Vppofe  that  4  ®  C  be  4  triangle: 

&  let  one.  of  )  fides jfierof ,  namt*  p 
\ly  M C he prqducedMto  the  point  ~ 

IDfThen  I fay,  that  the  outwarde 
angle  A  CD }ts greater  then  any  one  of  J  ttoo 
inward  and  opposite  angles  fat  is  fen  the 
angle  CB  A,or  then  the  angle  B  A  CJDeuide 
the  line  A  C (by  the  io .propofition)  into  two 
equall  partes  fin  the  point  E.Anddra'fr  a  line 
from  the  point  B  to  the  point  E:  And  (by  the 
2.  petition)  extend  BE  to  the  point  E.jfnd 
(by  the  z,  propofition)  onto  the  line  B  Eput 
an  equall  Ime  E  F.  And(by  the  fir fl  petition) 
dra'toa  line  from  F  to  C.and  (by  the  z.petici * 
on)  extend  the  line  AC  to  the  point  G.MoTo  forafmuch  as  the  line  A  E, is  equall 
to  the  line  E  C,  and  B  E  is  equall  to  E  E:  therfore  thefe  nvo  fide  s  A  Eand  E  B, 
are  equall  to  thefe  neo  fides  C  £  and  E  F,  the  one  to  the  other,  and  the  angle  ft 
E  Bfis  (by  the  i  5,  propofition)  equall  to  the  angle  F EC,  for  they  are  hedan* 
gles:  therefore  (by  the  4.  propofition)  the  bafeA  B  is  equall  to  the  bafe  F  C: 
And  the  triangle  ABEis  equall  to  the  triangle  F  E  C:and  the  other  angles  re* 
mayning  are  equall  to  the  other  angles  remaymngfe one  to  the  other  fnder 
Tfihicb  arefubtended  equall  fides,  VEherefore  the  angle  B  AE  is  equall  to  the 
angle  E  C  F„  But  the  angle  E  C  ID  Js  greater  then  the  angle  E  C  FyVherefore 
the  angle  A  C  £)}is  greater  then  the  angle  B  A  C.  In  like  fort  alfo  if  the  line  B 
C  be  deuidedinto  two  equall  partes , may  it  be  prouedfat  the  angle  B  C  &3tbat 
is,  the  angle  A  GAD,  is  greater  then  the  angle  ji  B  C.VVhenfoeuer  therfore  in 
any  triang  ley  the  line  of  one  fide  is  dr  awen forth  in  length :  the  outward  angle 
fbalbe greater  then  any  one  o  f  the  ttao  inward  and  oppofite  angles :  ’tobich  l#a$ 
required  to  be  demonstrated. 

jin  other  demonjlration  after  Telitarius, 

Suppofe  that  the  triangle  geuen  be  A  *B  C,  Whofe  fide  AB  let  be  produced  vnto 

H*j*  the 


Cgnftr&aim. 


Demsnftrdtiem. 


An  other  D&* 
wonfiratian  af-* 
ter  PslitHriut, 


tire  point  D.  Then  I  fay,  that  the  angle  D  B  Ck  greater  then  either  of  the  angles  BAG 
an dA  CB* For  forafmuch  as  the  two  lines,  A  C and  B  C do  concurre  in  the  point  C,and 
ypon  them  faileth  the  line  A  B:  therefore  (by  the  conuerfe  of  the  firftpeticionjthe  two 
inward  angles  on  one  and  the  felfe  fame  fide,are  leffe 
then  two  right  angles.Wherefore  the  angles  A  B  Cand 
CAB  are  leffe  then  two  right  angles :  but  the  angles  A 
B  C and  DBC arc  (by  the  1.3  proportion)  equal  to  two 
fight  angles .  Wherefore  the  two  angles  A  B  C and  2?  B 
C  are  greater  then  the  two  angles  «A  B  Cand  B  AC. 

Wherfore  taking  away  the  angle  zA  B  C,which  i s  com¬ 
mon  to  them  both,  there  {hall,  be  left  the  angle  B)B  C 
greater  then  the  angle  B  AC .  And  by  tlie  fame  reafon, 
forafmuch  as  the  two  lines  B  A  and  CA  concurre  in  the 
point  A, and  vppon  them  faileth  the  right  line  C2?,the 
two  inward  angles  ABC  and  ACB  are  leffe  then  two 
right  angles.But  the  angles  zA  B  Cand  B)  B  C are  equall  to  two  rightangles.  Wherfore  - 
the  two  angles  ABC  and  D  B  C,are  greater  then  the  two  angles  AB  C  &.AC  B.  Wher¬ 
fore  taking  away  the  angle  zA  B  C, which  is  common  to  them  both,  there  fhal  remaine 
the  angle  DBC  greater  then  the  angle  ACB-.  whichwasrequiredtobeproued. 

Here  is  to  be  noted,that  when  the  fide  of  a  triangle  is  drawen  forth,  the  angle 
ofthe  triangle  which  is  nexttbe  outward  angle,  is  called  an  angle  in  order  vnto 
it:  and  the  other  two  angles  of  the  triangle  are  called  oppofite  angles  vnto  it* 

UCotniUrj  Of  this  Propofition  followeth  this  Corrolkry ,  that  ft  is  not  poffible  that  from  one  & 

of  this  the  felfe  fame  point  fhould  be  drawen  to  one  and  the  felfe  fame  right  line,  three  equall 

frspojsuot*.  right  lines.  For  from  one  point,  namely,  A,  if  it  be 
poffible,let  there  be  drawen  vnto  the  right  line  BB), 
thefe  three  equall  right  lines  zA B,A  C,8cA  D. And 
forafmuch  as  A  Bis  equall  to  A  C,the  angles  at  the 
bafe  are  (by  the  fifth  propofition Jequall. Wherfore 
the  angle  AB  Cis  equal  to  the  angle  ^  C.Z? .Agayne 
forafmuch  as  A  Bis  equall  to  A  D,  the  angle  ABB) 
is  (by  the  fame)  equall  to  the  angle  ADB  :  but  the 
angle  A  B  Cwas  equall  to  the  angle  ACB.  Where¬ 
fore  the  angle  ACB  is  equall  to  the  angle  tAB)B : 
namely,the  outward  angle  to  the  inwarde  &  oppo¬ 
site  angle:  whichisimpoffible.Wherforefrom  one 
and  the  felfe  fame  point,can  not  be  drawn  to  one  &  B 

the  felfe  fame  tight  line  three  equall  right  lynes: 
which  was  required  to  be  proued. 


c 


A&  other  Cor - 
r&lLsrj follo¬ 
wing  alfa  of  the 

ftme. 


By  this  propofition alfo  may  this  be  demonftratcd,that  ifaright  line  felling 
\'pon  two  right  lines, do  make  the  outward  angle  equall  to  the  inward  and  oppo- 
fice  angle,thofe  right  lines  fhallnot  make  a  triangle, neither  flial  they  concurre. 
For  otherwife  one  St  the  felfe  fame  angle  fliould  be  both  greater,and  alfo  equal: 
which  is  impoffible*As  for  example. 


Suppofe  that  there  be  two  right  lines  AB  and  C  D,  and  vpon  them  let  the  right  line 
B  E  fall,  making  the  angles  ABB)  and  CDE  equall.Then  I  fay,that  the  right  lines  AB 
andCDfhallnotconcurre.Foriftheyconcurre,theforefaide  angles  abidyng  equall, 
name!y,the  angles  C‘Z>£  and  >2.5  D  :  Then  forafmuch  as  the  angle  CDE  is  the  out¬ 
ward  angle  it  is  of  neceffitie  greater  then  the  inward  and  oppofite  angle,  &it  is  alfo  e- 
qual  vnto  it :  which  is  impoffible.  Wherfore  if  the  feid  lines  cocurre,the  fhal  not  the  an¬ 
gles  remayne  equalgbut  the  angle  at  the  point  B)  ihall  be  encrcafed.For  whether  zA  B 

abiding 


ofjzucmes  memenm.  FoLz6« 

abidirt^fixedyou  fuppofe  the  line  CD  to  be  moued 
vnto  itjfo  chat  they  concurre, the  fpace  and  diftaiice 
in  the  angle  will  be  greater :  for  liow  much  more  C 
X>approcheth  to  eABJo  much  farther  of  goethjt 
from  D  E.  Or  whether  C D  abiding  fixed,you  ima¬ 
gine  the  line  tA  B  to  be  moued  vnto  it,  fo  that  they 
concurre,the  angle  °X  B  ‘D  will  be  leffe,  for  there¬ 
with  all  it  commeth.ne  re  vnto  the  lines  CD  &BD, 

Or  whether  you  imagine  either  of  them  to  be  mo¬ 
ued  the  one  to  the  other,  you  ihall  findc  that  the 
line  sA  B  camming  neere  to  C  23  ,  maketh  the 
angled B  E  ldTe,and  C D  going  farther  from  DE 
by  reafon  of  his  motion  to  the  line  B  ^maketh  the 
angle  CDE to  increafe.  Wherefore  it  followeth  of 

neceffitie,thatif  it  be  a  triang!e,and  that  the  right  lines  *A  B  and  C D  do  concnrre,the 
outward  angle  alfo  {hall  be  greater  then  theinward  and  oppofite  angle.  For  either  the 
inward  and  oppofite  angle  abiding  fixed,the  outward  is  increafed:  or  the  outwardea- 
biding  fixed,the  inward  and  oppofite  is  diminifhed:  or  els  both  of  them  being  moued 
till  they  concurre,the  inwarde  is  diminilhed,  and  the  outwarde  is  more  inereafed: And 
the  caufe  hereof  is  the  motion  of  the  right  lines  the  one  tending  to  that  parte  where  it 
diminifheth  the  inwarde  angle,  the  other  tending  to  that  part  where  it  inereafeth  the 
outward  angle. 


B 

A 

C 

D 

E 

be  io.  Fheoreme.  cTheiy9  Trogojition* 

<^n  euery  triangle gwo  angle s>  Vahich  mo  foemr  be  taken^  art 
lejie  then  Wo  righ  t  angles. 


2$ ^PP(sfe  ‘that 'A  B  C  be  a 
gy 1 1 f triangle,  Then  I [aye  that 
^  two  angles  of  the  fay  d  tri* 

mm  -  -  ■  i a 


Icjkihi  mo-. right  angles 
!  &xtend(  by  the  Z-peticio) 

-A  the  line  BC\  to  the  point 

''  v  •  *  £  ~  ’  'Q 

ID.  And  fora  [much  as  (by  the  propoftion going 

before)  the  outward  angle  of  the  triangle  A  BC^  namely  the  amle  A  CD  is 
greater  then  the  inward  and  oppofite  angle  A  B  C: put  the  angle ACB comma, 
to  them  both  therefore  the  angles  AC  D  and  AC  B  'are greater  then  the  an * 
gksABCand B  C  A  .But  (by:  the  13  propoftwiftheangies  A C  -Dana  AC  B 
are  equal!  to  mo  right  angles ,  VC ’here} ore  the  angles  ABC  and  B  CAare 
left  then  m/rigU  angles .  hi  like  fort  alfo  maygpe  prone  y  that  the  angles  B  A 
Cand  AC  £  are  left  then  mo  right- angles  l  andaljo  that  the  angles ,C  AB  <(3* 
ABC  are  left  'then.  mo  right  angles.  Wherefore  in  euery  triqngftmo  cm* 
gles  Tthicb  mo  foeuer  be  taken ^re  lejfe  then  mo  right  angiesmhsch  Iras  re* 
quindtoheproMd,  "f  '  V  - 

. H.ii.  '  Tins 


Gtnjtrfi8i$%a 

DemenftrutmS, 


■A.. 


ThefirftHookp 

This  may  alfo  be  demonftrated  without  thchelpe  ofthe  former  proportion, 
by  the  conuerfe  ofthe  fifth  petition,  and  by  the  i$.propoficion  as  you  faw  was 
done  in  the  former  after  Pclittrm. 


It  may  alfobe  demonftrated  without  producing  any  of  thefides  ofthe  tri* 
angle, after  this  maner. 

Another  demt-  '  SuPPofe  that  there  a  be  triangle  ABC  And.  in  the  fide  B  C  take  apointatalladuen- 

ftrxtion  tnuen-  turessand  let  the  fame  be  D,  and  draw  the  line  A  D.  And  forafmuch  as  in  thetriangle 
sedbj  Proclus .  A  ^  D*  the  fide  B  D  is  produced,  therefore  (  by  the  former  proposition )  the  outward 
angle  D  C,is  greater  then  the  inward  and  oppofitc  angle  A  B  D. Agayne  forafmuch 

as  in  the  triangle  A'D  C,  the  fyde  CD  is  produced, thereforefby  the  fame)the  outward 
angle  A  D  B,is  greater  then  the  inwarde  and  oppofitc 
angle  D  :  but  the  angles  at  the  point  Darcequall 

to  two  right  anglesfby  the  xj.propofition:  jwherforc 
the  angles  A  BCznd  A  CB  are  lefle  then  two  right  an¬ 
gles.  And  by  the  fame  reafon  may  we  proue  that  the  an 
gles  B  AC  and  B  C  A  are lefle  then  two  right  angles, if 
we  take  a  poynt  in  the  line  <sA  C,  and  draw  a  right  line 
from  it  to  the  point  B  :  and  fo  alfo  may  it  be  proued 
that  the  angles  CtAB  and  *A BC  are  lefle  the  two 
ryght  angles  ,if  there  be  taken  in  thelyne^^apoint, 
and  from  it  be  a  line  drawen  to  the  point  C. 


Corrolltry 
following  this 
Progojitson, 


ConrtrHthon . 


-v 


By  this  proportion  alfo  maybe  proued  this  Corrollary,  that  from  oneand 
the  feite  fame  point  to  one  and  the  felfe  fame  right  line, can  not  be  drawen  two 
perpendicular  lines. 

For  ifitbepoflible,  from  the  point  ^,Iet  there  be  drawen 
vnto  the  right  lrae2?C,two  perpendicular  lines  AB^  and  ACz 
wherefore  the  angles  A  B  Cand  CB  are  right  angles.  But 
forafmuch  as  A  B  Cisa  triangle,  therefore  any  two  angles  ther- 
of  are  (by  this  propofition )  lefle  then  two  right  angles.Where- 
fore  the  angles' Af  J?  C and  A  CB  are  lefle  then  two  right  angles: 
but  they  are  alfo  equall  to  two  right  angles.by  reafon  A  B  and 
<tA  C  arc  perpendicular  lines  vpon  B  C:  which  is  impofllble. 

Wherefore  from  one  and  the  felfe  fame  point  cannot  be  drawc 
to  one  and  the  felfe  fame  line  two  perpendicular  lines  j  which 
was  required  to  be  proued. 

The  ii.  Theoreme .  The  i  SJPropofition. 

In  euery  triangle ,  to  the  greater  fide  is  fubtended  the  grea* 


Vppofe  that  ABC  be  a 
triangle ,  bauing  the  fide  A 
C greater  then  the  fide  A 
B.  Then  l  jay  that  the  an* 
vie  A'BC  is  greater  then 
the  angle  B  QA .  For  for* 
afmuch  as  A  d  is  greater 
the  A  Bfput(by  the  3 .pro - 
pojition)  Vnto  A  B  an  equall  line  A  &  •  And  (by 


of  buckoes  blementes .  roLij* 

the  firfl  petition)  draw  a  tine  from  the  point  B  to  the  point  (D.  And  forajmnch 
as  the  outward  angle  of  the  triangle  IDBC,  namely }  the  angle  aB>B  isgrea* 
ter  then  the  inittardand  oppo/ite  angle  VCB  (by  the  16.  propo/ition))but  (by 
the )-.  propofition)  the  angle  AB)[ B  is  e quail  to  the  angle  A  B  B)Jor  the Jyde 
A  B  is  equal!  to  the  jyde  A  ID;  therefore  the  angle  A  BID  is  greater  then  the 
angle  ACB.  Wherefore  the  angle  ABC  is  much  greater  then  the  angle  A  C 
B.  Wherefore  in  euery  triangle,  to  the  greater  Jyde  isfubt  ended  the  greater 
angle:  trhich  Snas  required  to  be  proued , 

You  may  alfo  proue  the  angle  at  thepoint  B  greater  then  the  angle  at  the  point  C 
(the  fide  A  C being  greater  then  ti  elide  AB Jif from  the  line  ACyou  cut  of  a  linee- 
quall  to  the  line  sA  2?,beginning  at  the  point  C, as  before  you  beganne  at  the  point  Ax 
and  that  after  this  manner.  Let  the  line  D  C  be  equall  to  the  line 
*A  B  and  draw  the  line  B  D:  and  produce  <iA  B  to  the  point  E :  A 
and  put  the  line  "B  E  equal  to  the  line  A  D.  Wherefore  the  whole 
line  AEis  equall  to  the  whole  line  A  C:draw  aline  from  E  to  C, 

And  forafmuch  as  AEis  equal  to  A  C,  therfore  the  angle  A  E  C 
is  alfo  equall  to  the  angle  A  fE  ("by  the  5. propofition.’)  but  the 
angle  ABCis  greater  then  the  angle  cA  E  C.For  one  of  the  fides 
of  the  triangle  CB  E,  namely,  the  fide  B  E  is  produced  ,  and  fo 
the  outward  angle  <tA  B  C  is  greater  then  the  inward  and  oppo- 
fite  B  EC  (by  the  1 6  propofition.-)wherefore  the  angle  c A  BC  is  & 
much  greater  then  the  angle  &A  C B  r  which  was  required  to  be 
proued. 

Notethat  that  which  is  here  fpoken  in  this  propofiti. 
bn* is  to  be  vnderftanded  in  one  and  the  felffame  triangle.  For  itispoffiblcthat 
one  and  the  felfe  fame  angle  may  be  fubtended  ofa  greaterline,and  ofa  Idle  line: 
and  one  and  the  felfe  fame  right  line  may  fubtend  a  greater  angle ,  and  a  ldfe  an- 
gl e. As  for  example , 

Suppofe  that  there  be  an  Ifofceles  triangle  AB  C,  & 
in  the  fide  AB  take  the  point  D  atall  aduentures:  &  fro 
the  line  A  C  cut  of(by  the  5  .propofition)  the  lyne  A  E 
equall  to  the  line  AD.  And  draw  a  right  line  from  D  to 
.E.Wherfore  the  right  lines  DE  and  BC  do  fubtend  the 
angle  at  the  point  A,  &  of  them  the  one  is  greater,  and 
the  other lefle.  And  after  the  felfe  fame  manner  a  man 
may  putinfinite  right  lines  greater  &  lefle,  fubtending 
the  angle  at  the  point  A, 

Agayne  fuppofe  that  ABC  be  an  Ifofceles  triangle. 

And  let  B  C  be  lefle  then  either  of  the  lines  BA  and  AC . 

And  vpon  B  C deferibe  (by  the  firfl: Jan  equilatcr  trian¬ 
gle  i?  C D.And  drawaline  from  A  to  2);  and  produce  it 
to  the  point  E.  And  forafmuch  as  in  the  triangle  A  B  A 

X>,the  outward  angle  B  D  E,is  greater  rhen  the  inward 
&  oppofxte  angle  BAD  (by  the  i  ^.propofition  )And  by 
the  fame  in  the  triagle  ACD,tbe  outward  angle  CDE,is 
greater  then  the  inward  &r  oppofiteangleCAZ>:ther- 
fore  the  whole  angle  BDCis  greater  the  the  whole  an¬ 
gle  B  AC .And  one  and  the  felfe  fame  right  line  fubten- 
deth  both  thefe  angles,namelv,the  greater  angle  &  the 
lefle.  And  itis  alio  proued, thatgreatcr  rightlines 
Sc  lefle  fnbtende  one  and  the  icIfefamcangle,Bntm 

K»uj*  one 


Dcmonfratk#* 


An  other  da- 
menftrntnn  af¬ 
ter  Prophjrms* 


That  which  ti 
fpoken  to  thus 
Props fisen  is  re 
be  Vnderftanded 
to  one  and  the 
felfe  fame  tri¬ 
angle. 


Dtmwjlratsott 

leading  to  an 
impoffibiliti-e. 


This  fropofition 

is  A  c  Conner  ft 
ibe former-. 


9 An  Affumpt  is 
a.  Propoftion  tn 
kya  of  neceffitie 
to  the  helps  of  a 
demon f ration, 
the  certainty 
te  here  of  ss  not 
Jo  plabie,  and 
■thsrfore  nedeth 
st  felfefrfl  to  be 
d-cmossfirated. 
Art  affumpt  pu-t 
by  Proclnsfor 
S  he  demonflra- 
tien  of  this  Pro- 
psfstssn. 


\  cThejtrBcBoo%e 

<3neandthc  fclfe  fame  triangle  one  right  lincfubtendeth  one  angle. and  the  great 
zn^the great  angleaand  the lcITc  the  ldTe,as  it  was  proued 

ThenfiTheoreme.  The  ipfPropoJilion* 
fn  euery  triangle ,  vnder  the  greater  angle  ujubtended  the 
‘greater  fide* 

Appofe  that  A  B  Cbea  triangle,  hauyng 
t be  angle  A  B  C greater  then  the  angle  B 
C  AT  ben  I fay  that  the  fide  AL  is  greater 
then y  fide  A  BT 'or  if riot. fife  the fide  AC  is  ether 
equal  toy  fide  AByr  els  it  is  lejfie  theitXbe fide  A 
C  is  not  equal  to  y  fide  ABfior  then( by  the  $,pro* 
pofition  )y  angle  ABC  fhould  beequall  to  the  an * 
gle  A  C  B:  but  (by fufipofitio)  it  is  not,  Where* 
fore  the  fide  A  C  is  not  equal!  to  the  fide  A  B.  And 
the  fide  A  C  can  not  be  lefie  then  the  fide  A3^  for  then  the  angle  A  B  Cfhoulde 
be  lejfie  then  the  angle  A  C  B(by  the  prop ofitim  nextgoyng  before).  But  (by 
flip  pofition  it  is  not)  Wherefore  the  fide  AC  is  not  lefie  then  the  fide  AB„ 
Wherefore  the  fide  A  C  is  greater  then  the  fide  A  B.  VAherefore  in  euery  tri* 
angle  r  'vndpr  the  greater  angle  is  fnbtended  the  greater fide :  'tohich  Taasreqm* 
red tg  be  demon fl rated.  . 

-  isi  r  5  Da 

This  propofition  is  the  conuerfeof  thepropofition  next  goingbeforc.VV,^ler-, 
fore  as  you  fee.* that  which  was  thecondufion  in'theforroer,is  m  this  the  fuppo- 
iitionjor  thing  geuemand  that  which  was  there  the  thing  geuen,is  here  the  thing 
required  or  concluiion.  And  it  is  proued  by  ah  argument  leading  to  an  impoffis 
biikieyis  commonly  all  conucrfes  are. ' 

Twins,  demon  jlrateth  th  is  proportion  after  an  other  way :  butfirftheputteth 
this ^AiTumpt following*  -,n.  ,  ;.q..y  . 


:A1 


If  an  angle  of  a  triangle  be  deuide  din  to  ttyo  eefnallpartes,  and  if  the  HneXvhich  dettideth  it  being 
dradren  to  the  baff  do  deuide  the  fame  into  t\ho  vneefuaH  partes :  the fides  'd’hich  contaynt  that  angle 
jhalbe  vneqttaH,  and  that  fiialbe  the  greater  fide,  tyhicbfalleth  on  the  grater  fide  of  the  bafc,andtha£ 
the  lefie  whtcbftHeth  on  the  lefie fide  of  the  baje.  M  ;  -  \ 

Cf  y  v  .  '  ••  -xilJ  ✓ill  n: ....  *•>  rA.  ,  ■ .  i  isVij 

Suppofe pA B  C  to  be  a  triangle,and(by  the  p .  proportion)  deuide  the  angle  at  the 
point  A,m\p  two  equall  partes,by  the  rightfitfe  A  jD.  And  let  the  line ^tD  deuide  the 
bafe  B  Quito  two  vnequailpartes^  andle.t  thbpart  CD  be  greater,  then  the  parteiMfc. 
Then  I  fay  .that  the  fide  AC  is  greater  thenthe  fide  B.  Produce  the  fine  A  D  to  the 
point  Qand(by  the  third  Jput-the  line  D  i? ’equal!  to  the  fine  D  A.  And  forafinuch  as 
DC  is  by  fuppoikion  greater  then  DA,  put  (by  the  3  .propofitionjD  Fequal  to  ADS 
and  draw  a  line  fro  E  to  Qand  produeeitto  thepoint  G ,  Now  forafinuch  as  .^Dise* 
quali  to  ED  and  D  A  is  equall  to  D  Ft  therfore  in  the  two  triangles  AB  D.and  EFD* 
two.fides  of  the  one  are  .equal!  to  two  fides  of  the  other ,  eche  to  his  correfp.ondenj 
fide’;  and  ( by  the  1 5 ■.  proportion)  they  contayae  equall  angle$5  nameiy3  the  hed  ari-- 


ofEucUdes  Elementes.  Fo!.i2* 


gies  :  wherfore  f  by  the  fourth  propor¬ 
tion  )  the  bafe  B  A  is  equall  to  the  bale 
£  F: and  the  angle  D  E  F  is  equall  to  the 
angle  D  A  B.  But  the  angle  DAG  is  by 
conftru&ion  equall  to  the  fame  angle  D 
zA B:  wherefore  (by  the  firft  common 
fentence )  the  angles  EA(j  and  AE  G 
are  equall.  Wherefore  (by  the  6.  propo- 
fion  )  the  fide  AG  is  equall  to  the  fide  E 
(7,Wherfore  the  fide  A  Cis  greater  then 
the  fide  £  G .  Wherefore  it  is  much  grea¬ 
ter  then  the  fide££.Butthe  fide  ££i$ 
equall  to  the  fide  A  3,  as  it  hath  bene 
proued. Wherefore  the  fide  A  Cis  grea¬ 
ter  then  the  fide  A  £j  which  was  requi¬ 
red  to  be  proued. 


This  aifumptbeing  put,this  Proportion  isofProclus  thus  dcmonftrated. 


Suppofe  A  3  Cto  be  a  triangle,  hauing  hisangle  at  the  point  £  greater  then  the  an 
gle  at  the  point  C,  Then  I  fay  that  the  fide  A  Cis  greater  then  the  fide  AB.Deuide  the 
line  B  C  into  two  equall  partes  in  the  point  £>,and  draw  a  line  from  A  to£>.  And  pro¬ 
duce  the  line  A  D  to  the  point  £ :  and  put  the  line  D  £  equall  to  the  line  A  Z>,and  draw 
alinefrom£to£.Nowforafmuchas££>  is  equall  to  DC,  and  AD  is  equall  to  DE 
therefore  in  the  two  triangles  A D  C  and  B  DE,  two 
fides  of  the  one  are  equall  to  two  fides  of  the  other,  ech 
to  his  correfpondent  fide,  and  they  containe  equall  an¬ 
gles  (by  the  i  propofition). -wherefore  (by  the  fourth 
proportion)  the  bafe££  is  equall  to  the  bafe  AC,  and 
the  angle  DB  Eis  equal  to  the  angle  at  the  point  f^De- 
uide  alfo  th’angle  ABE  into  two  equal  parts  by  the  line 
B  F:  wherfore  the  line  EE  is  greater  then  the  line  F  A. 

And  forafutuch  as  in  the  triangle  A  B  £,the  angle  at  the 
point  B  is  deuided  into  two  equall  partes  by  the  right 
line  B  F,  and  the  line  £  £  is  greater  then  the  line  A  F : 
therefore  by  the  former  AJfumpt  the  fide  £  £  is  greater 
then  the  fide  BA:  but  the  line  £  £  is  equall  to  the  line  A  C.  Wherfore  the  fydcAC  is 
greater  then  the  fide  A B:  which  was  required  to  be  proued. 

The  i i.lhenreme.  The  20.  Tropofition . 

In  euery  triangle  tier 0  fides,  which  two  fides  fioeuer  be  taken, 
are  greater  then  the  fide  remajning. 


A 


Vuppofe  that  ABC  be  a 
triangle.  Then  IJay  that 
t^o fides  of  the  triangle  A 
B  C,  which  two  fides  foe * 
uer  be  taken ,  are  greater : 
'then  the  fide  remayning 
^  j  that  is,  the  fides  B  A  and 
~  AC  are  greater  then  the 
Bdifi  fit 


An  ether  de- 
monff ration  afs 
ter  Preelm. 


Gtmtfrucffc >#. 
&emsvj?r£ts$n. 


\A»  ether  dema  ■* 
flrttoton  with¬ 
out  producing 
sits  of  the  ftdcs . 


jht  ether  De- 

monfimtkn9 


*  TTjeJirH^oofy 

ftd c  B  C:and  the  (Ides  A  B  and  B  C  then  the  fide  A  C:  and  the fides  A  C  and  B  C 
then  the  fide  B  AfProduce  (by  the  2,peticion)the  line  ©  A  to  the  point  D,And 
(by  the  third  propo/ition finto  the  line  A  C  put  an  equall  line  A  3);  and  dram 
a  line  from  the  point  D  tothepointe  C.  And  forafi 
much  as  the  line  DA  is  equall  to  the  line  A  Cohere* 
jore(by  the  ^,propoftion)the  angle  A3)  C  js  equall 
to  the  angle  A  C  D.But  the  angle  BCD  is  greater 
then  t  he  angle  A  C  D,t  here  fore  the  angle  BCD  is 
greater  then  the  angle  ADC,  And  farafmnch  as  D 
CB  is  a  triangle y  bauing  the  angle  BCD  greater 
then  the  angle  A  D  C,but(by  the  i$,propofitton)viu 
der  the  greater  angle  is  fubtended  the  greater  fide: 

Tt> herfore  D  B  is  greater  then  B  C.  But  the  lineD 
B  is  equall  to  the  lines  A  Band  A  C(for  the  line  JD  is  equall  to  the  line  A  C) 
If  herfore  the files  B  A  and  A  C,  are  greater  then  the  fide  BC.And  in  like  forte 
may  foe  prone  f  hat  the  fides  A  Band  B  C  are  greater  then  the  fide  A  C:&  that 
t  he fides  B  C  and  C  A  are  greater  then  the  fide  A  B  VVherfore  in  euery  t  riant 
gle  tlfo  (idesftohich  two  tides  focuer  be  taken, are  greater  then  the fide  remay * 
ning\ ifhich  IVas  required  to  be  demonfir ated. 

This  Proposition  may  alfobe  detnonftrated  withoutproducing  any  of  the 
fides,  alter  this  mancr. 

Suppofe  ABCt  o  be  triangle.  Then  I  fay,  that  the  two  fides  AB  and  A  Care  grea¬ 
ter  then  the  fide  B  C:  deuide  the  angle  at  the  point  A  (by  the  9.  propofition)into  two 
equal!  partes  by  the  right  line  <tA  E.  Andforafinuch  as  in  the  triangle^  B  £,theoufr* 
ward  angled  EC  is  greater  then  the  angle#  ex/  E  (by 
the  i<5propofition),and  the  angle  BAEis  put  to  be 
equall  to  the  angle  E  A  (^therefore  the  fide  AC  is  grea¬ 
ter  then  the  fide  (fE,  And  by  the  fame  reafon  the  fide 
A  Bis  greater  the  the  fide  B  £JFor  in  the  triangle  AEC 
the  outward  angle  AEBtis  greater  then  the  angle  CA 
<£\that  is  then  the  angle  Wherelorealfb  the  fide 

AB  is  greater  then  the  fide  BE.  Wherfore  the  fides  A'B 
and  A  Care  greater  then  the  whole  fide  B  C.  And  after 
the  fame  maner  may  you  proue  touching  the  other 
fides  alfo. 


The  fame  may  y  etalfo  be  demonfi:  rated  an  other  way. 


Suppofe  A  B  C  to  be  a  triangle.  Now  if  A  B  C  be  an 
equilater  triangle,  then  without  doubt  any  two  fides 
thereof  are  greater  then  the  third.  For  the  three  fides 
being  equallany  two  fides  of  them  are  double  to  the 
third.But  if  it  be  an  Ifofceles  triangle,  either  the  bafe  is 
lefie  then  either  of  the  equall  fides  or  it  is  greater.  If  the 
bafe  be  Iefle,  then  againe  two  of  them  arc  greater  then 
thethirde,  butif  the  bafe  be  greater  (let  BC  being  the 
bafe  of  the  Ifofceles  triangle  ABC  be  greater  the  either 
of  the  fides  AB  Sc  AC  and  from  it  cut  of  (by  the  j.pro- 

‘  pofition 


Polity* 

portion )  &  line  equall  to  any  one  of  the  other  fides,  whiche  let  bee  f-E^and  dtawe 
a  line  from  A to  E. And  forafmuch  as  in  the  triangle  A  E  B, the  angle  A  E  C is  an  out¬ 
ward  angles  therefore  it  is  greater  then  the.  angle  B  AE  (by  the  1 6.  proportion)  .And 
by  the  fame  reafomthe  angle  AEB  is  greater  then  the  angle  CAE .  Wherefore  the  an¬ 
gles  at  the  poiht  E  ate  greater  theh  the  whole  angle  at  the  pointe  A.  But  the  angle  B  E 
A  is  equal  to  the  angle  B  A  E  (by  the  f  *  propofition)  for  Ce B  is  put  to  be  equall  to  B : 
£.  Wherefore  the  angle  remayning  A  Efts  greater  then  the  angle  CAE,  Whereforei’ 
alfo  the  fide  A  C  is  greater  then  the  fide  E  (*.  But  the  fide  AB  is  equall  to  the  fide  2?  £* 
Wherefore  the  fides  A B  and  Care  greater  then  the  fide  B  C, 

But  if  the  triangle  A  B  C  be  a  Scalenum,  let  the  fide 
A  B  be  the  greateft,  and  let  A  C be  the  meane,  and  B  C 
the  leaft.  Wherefore  the  greateft  fide  being  added  tip  a-  ^ 

ny  one  of  the  two  fides  mull  nedes  be  greater  then  the 
third.  For ofit  felfe  it  is  greater  then  any  of  them .  But 
ifAB  being  the  greateft,you  would  proue  the  fides  AC  X  /  \\ 

and  CB  to  be  greater  then  it.  Then  as  you  did  in  the  I-  /  /  \\  » 

foceles  triangle,  cut  of  from  the  greateft  a  line  equall  to 
one  of  them,  and  from  the  point  Cto  the  point  of  the 
interfedtion  draw  a  right  line,and  reafon  as  you  did  be 
fore  by  the  outward  angles  of  the  triangle,andyou  ihal 
haueyourpurpofe. 

This  propofition  may  yet  moreouerbedemonftratedby  an  argument  lea¬ 
ding  to  an  abfurditie,  and  that  after  this  manner. 

Suppofe  ABC  to  be  a  triangle,  then  T  fay  that  the 
fides  A  B  and  A  C,  are  greater  then  the  fide  B  C.  For  if 
they  be  not  greater,they  are  either  equall  or  leffe.  Firft 
let  them  be  equall,and  from  the  line  B  C  cut  of  the  line 
B  E  equall  to  the  line  A  B( by  the  ^propofition)  wher- 
fore  the  refidue  E  Cis  equal!  to  A  C.Now  forafmuch  as 
A  B  is  equall  to  BE  they  fubtend  equall  angles.  Like- 
wife  forafmuch  as  «xr  Cis  equall  to  CE  they  fubtend  e- 
qual  angles.Wherfore  the  angles  which  are  at  the  point 
fare  equall  to  the  angle  swhiche  are  at  the  pointed, 
which  is  impoftiblef  by  the  1 6. propofition). 

But  now  let  the  fides  A B  and  AC  beleffethen  the 
fide  B  C ,and  from  the  line  B  C  cut  of  ( by  the  3  .propofi- 
tion)the  line  B  D  equall  to  the  line  A  B ,and  likewife  fr5 
the  fame  line  B  C  cut  of  the  line  CE  equall  to  the  line  A 
C.  And  forafmuch  as  is  equall  to  f  O,  the  an<de  B 
2>  ^  alfo  is  equall  to  the  angle  BAD  (by  the  fifthpro- 
pofition).Againe  forafmuch  as  zsf  C  is  equall  to  CE 
therefore(bythe  fame)  the  angle  CEAis  equall  to  the 
angle  £  *A  C .  Wherefore  thefe  two  angles  B  D  A  and  C 
€  A  are  equall  to  thefe  two  angles  B*AD  and  £  ^4  C, 

Agayne  forafmuch  as  the  angle  EDA  is  the  OtitwardU 

angle  of  the  triangle  ADC,  therefore  it  is  greater  thenB  D  £  C 

theanglef^C  Foritisgreaterthentheangle‘Z)^Cfbvthe  id.nronofmnn  ) 

by  the  fit  me  reafon/orafimuch  as  CE^tU  tSc  the  outwaS  angle of 
^  therefore  it  is  greater  then  the  angle*  A D  (for  it  is  greater  then  the  an S!/e  ) 
Wherfore  the  ang  es  *  DAandCE  A  are  greater  then  fhe  two  angfej  aHaeA 
.CBut  theywerealfo  proued  equal!  vnto  them  .•  which  is  impoffible.  Wherefore  the 
M'.sAB  and  A  Cure  neither  equall  to  the  fide  *  C,  nor  leffe  thenit  but  neater  And 

foalfo  nuy  itbeprouedofthereft.-  7  ?  out  ^reate,.  Ana 

A 


-l  "i  -i .  ;!.t  « 


edif  cthef  deffto'* 
ftrttticn  leading 
SeAnAfurdiM 


ThefirflcBooh$ 

^tiriefe  demon-  ,  morehriefely  demonftrate  this  proportion  by  Carapanus 

f mtion  bj  the  definition  ofa  right  line,  which  as  we  hauebefore  declared  isthus:  A  right  line  is 
'definition 9f«  tbefomeft  extenfm or  Jrafygbt  that  is  dr  ftiay  be  from  one  point  to  another. SNhcdore.  any  one 
mght H»et  fide  p fa  triangle,  for  that  it  is  a  right  line  drawenfrom  fome  one  point  to  Come 

other  one  point, is  of  neceffitie  lliorter  then  the  other  twofides  drawen  from  and 
tpjthc  famepointes*-  , t  •'  •  -/ 

•3K  %Ll-  cufiOJilftOfoai  ■  V  sr--  7', >  ;  V..;  rt  OK.  1 

a aaftthfttget  Epicurus fuch  as  followed  himderided  this  propofition,  not  counting  it 
IfifcfJt ftllfht  ^ortV  to  be  added  m  the  number  of  proportions  of  Geometry  for  the  eafines 
»  thereof,  for  that  it  is  mam  fell  euen  to  thefenfe.  But  not  all  thinges  manifeft  to 
veafon  *nd  c-«-  fenfe,  are  ftraight  wayes  mam  fell;  to  reafon  and  vnderllanding-  Tt  pertay  neth  to 
Jerfttmiingt  one  that  is  a  teacher  of  Sciences ,  by  profe  and  demonftration  to  render  a  cer- 

tayneand  vndoubted  reafon,  why  itibappeareth  to  thefenfe:  and  in  thatonely 
confiffeth fcience. 

/  -  s  or  S:  •.  •.  'oJitottrtgtfBrn  •  ... .  j^oi 

Theu^ThcoreMe.  TheilfPropofition* 

fffrom  the  aides  of  one  of  the  fide  s  ofa  triangle,  he  drawn 
to  any  point  vpithin  the  fayde  triangle  two  right  lines*  thofe 
right  lines  fo  draWenfalbe  kjle  then  the  two  other  jides  of 
f  a  -  the  triangle,  but jhaUcont dine  the greater  angle. 

Vppofe  that  ABC  be  a 
triangletand  fro  the  endes 
oftheJide'B  C ^namely  fro 
the  point e s  3  and  C  y  let 
[there  be  drawen  Taithin  y 
j triangle  right  lines  3 

\D  and  CD  to  y  point  IX 
Then! fay ,  that  the  lines 
3  3)  and  C  ID  are  lejs’e  then  the  ether  jides  of  the 
triangle,  namely,  then  the  Jides  3y{  and  AC:md  that  the  angle  fbhich  they 
contayne,  namely,  3  DC,  is  greater  then  the  angle  3  AC.  Extend  (  by  the 
ficond  petition)  the  line  3D  to  the  point  Et  And  for af much  as ( by  the  Zo. pro* 
pojition)  in  euery  triangle  the  ttrofides  are  greater  then  the  fide  remayning% 
therefore  the  t'&o Jides  of  the  triangle  A3  E,  namely,  the Jides  A  3  and  A  Ey 
are  greater  then  the fide  E  3.  Tut  the  lineE  C  common  to  them  both.  Where < 
fore  the  lines  3  A  and  A  C, are  greater  then  the  lines  3  E  and  E  C.Againe  for * 
afrmb  as  ( by  the  famejin  the  triangle  C  E Dyhe  tlt>o Jides  C  E  and  E  D,  are 
greater  then  yfide  D  C,  puty  line  D  3  common  to  them  botht  Tbherforey  lines 
CE  and  E  B^are  greater  then  thelinesCDand D  3. 3ut  it  isproued  that  the 
Ones  3 A  and  AC^are  greater  then  the  lines  3  E and  ECj/Vhere fore  the  lines 
BA  and  A  C3are  much  greater  then  the  lines  3  D  and  D  C.  Agayneforafmuch 

\  as 


A 


fcf*  Fol^o* 

as  (by  the  1 6.  proportion)  in  ettery  triangle  ^theoutward  angle  is  greater  then 
the  inward  and  oppojite  angle  y  therefore  the  outward  angle  of  the  triangle  C 
IDE9  namely  fBB  Ctis greater  then  the  angle  CEB).  VV her e fore  alfo  (by  the 
fame)  the  outward  angle  of  the  triangle  A  B  E*  namely  9  the  angle  CEB  is 
greater  then  the  angle  B  AC.  But  it  is  prouedythat  the  angle  BBC  is  greater 
then  the  angle  C  EB.VVherfore  theangle  BCD  Cis  much  greater  then  the  ana 
gle  B  J  C .Where fore  if  from  the  endes  of  one  of  thefidesofa  triangkybe  dra* 
Wen  to  any  point  within  the  fay  de  triangle  tW o right lines  itho/e  right  lines fo 
drawn Jhalbelefte  then  the  two  other  fedes  of  the  triangle ,  but Jhallcontayne 
the  greater  angle :  which  was  required  to  be  demonflrated. 


.tone:" '< 


r.iuzn  IV :! 


In  this  propofition  is  exprefledjthac  the  two  right  lines  drawen  within  the 
triangle,haue  their  beginning  at  the  extremes  of  the  fide  ofth:  triangle*  Tor  fio 
the  one  extreme  of  the  fide  of  the  tr  langl  e,and  fro  ttifome  o  nepo  i  n  t  of  the  lame 
fide, may  be  drawen  two  right  lines  within  the  triangle, which  fliail  be  longer  the 
the  two  outwardlines:  which  i  s  wonderfull  and  feemeth  firaunge,  that  two  right 
lines  drawen  vpon  a  par  te  of  a  line,fiiould  be  greater  then  two  right  lines  drawen 
vpon  the  whole  lme.And  agay  ne  it  is  poffible  from  the  one  extreme  of  the  fide 
ofatriangle,andfrom  fome  one  point  ofthefamefide  to  drawe  two  right  lynes 
within  the  triangle  which  fiiall  contain^  an  anglelefic  then  the  angle  contayned 
yndcr  the  two  outward  lines*  3s"7V*wH  ' 


As  touching  the  firfi:  part* 

Suppofe  A  B  C  to  be  a  re&an- 
gle  triangle/whofe  right  angle  let 
be  at  the  point  B.  And  in  the  fide 
B  C  take  a  point  at  al  aduentures, 
which  let  be  D:  and  draw  a  right 
line  fro  AtoD  Wherfore  the  line 
AD  is  greater  then  the  line  AB 
fby  the  ip.propofitio)  From  the 
line  A  D  cut  of  (  by  the  thirde)  a 
lineequallto  the  line  A  B  which 
let  be D  E.  And  deuide  the  line  E 
Ain  to  two  equall  partes  in  the 
point  F(by  the  io. propofition ) 
And  draw  aline  fro  F  to  C.  Now 


-£r:;i  s 


"i  i  r.*1 


iaoi  -j'i 


: in  - 


forifmuch  zsAFCisz  triangle.thcrfore  the  lines  A  Fund  F  Care  greater  th?  r  i  •  „ 
j  ^ormerProP°htionJ.-but«^?' F  is  equal  ro  F-Eiarherforethe  rmktr  f 

and  F  ©are  grater  then  the  line  A  C. Arntthe  line  ©  B  is  equall  to  the  line!^  s" wh  E 
fore  the  right  lines  F  ©and  FD  are  greater  then  the  right  lines  j?  B^T/r  r‘ 

are  drawen  within  the  triangle  ABC,  the  one  from  one  extreme of 
Other  from  appint  in  the  fame  fide* C  ;  whiche  was  required  to  be  proued  *** 


A  ’  *r\  rf 

v.1  •» 


l  4  J 


Av* 


v*r>  * 


equall  fydes,  and  from  the  lyne  TJC  cuttcofalin  *  eauall  to  thr  itn.  t 

thirde  propofition)  whiche  le  t  bee  BV:  and  drawc  a  line  from  to©  ^ mid  in  the 

li*  line 


Thefirft'Bbo\ i 


Sine  tA  Drake  a  point  at  al  aducnturcs,  which 
let  be  £,  &drawalinefrom,Cto£.  Nowfor- 
afrnuch  as  the  line  is  equal  to  the lyne  S 

2>1therfore(bythefiftpropofition)the  angle 
B  AD  is  equall  to  the  angle BD  A.  And for- 
afmuch  as  in  the  triangle  E  D  Cthe  angle  ED 
Bis  an  outwarde angle,therefore  (by  the  id. 
proportion)  it  is  greater  then  the  inw  ard  and 
oppofiteangie  DEC,  Wherefore  the  angle  B 
AD  is  greater  then  the  angle  DEC,  Wherfore 
the  angle  B  AC  is  much  greater  then  the  an¬ 
gle  D  EC:  andtheanglcB  ACis  contained  of 
the  outward  right  lines,5  y^an:d  A  C?  and  the 
an  gl  eD  EC  is  contayned  of  the  inward  right  * 

lines  D  E  and  E  C:  which  was  required  to  be  proued. 


By  meancs  of  this  propesfi- 
tio n  alio  i  s  defer  ibed  that  ky  nd 
of  triangles, which  contayncth 
fourefides,  A  s  for  example, .this 
figu re  A  B  C.For  it  is  retained, 
ot  towel- fides  B  A, A  C,C  BjSC 
EB.But  ii;  hath  onely  three  a$-. 
gle  s,onc  a t.  the  p  o  in  t  B,an  other. 
atthepointA.andthe  third  at 
the  point  C.  VVhereforethis 
prefent  figure  ABC  is  a  qua- 
drilatertriangle:  whichofolde  B 
philofophcrs  hath  eucr  bene  counted  wonderful!, 

And  here  is  to  be  noted,  that  there  is  difference  bc- 
tweqe  a  three  fided  figure,  and  a  three  angled  figure. 

For  not  euer>  figure  hauing  three  angles  hath  alfo 
onely  three  fides5as  ir  is  platnc  to  fee  in  this  figure*, 

Likewife  it  is  not  all  on,a figure  to  haue  lower  fides, 
and  fower  angles,  Forafoure  fided  figure  may  hauc 
onely  threangles,as  in  the  former  figure:  andafoure 

angled  figure  may  hauc  fine  fides,  as  in  this  figure  to lo wing,  And  fo  of allothcc 
figures,  -  -  !  -:Vr  i'* 


’-it+UiU 


ofrtion • 


Ofthreright  lines, which  are  equall  to  tbre  right  lines  gem* 
to  makg  a  triangle . ! Butitbehoueth  two  ofthofe  lines, which 
mo/oeuer  be  taken, to  be  greater  then  the  third.  For  that  in 
mery  triangle  mo  fides,  which  mo fides foeuer  be  taken,  are 

grea* * 


greater  then  the  fide  remaining* 

Vppofe  that  the  three  right 
lines geue  be  A,B,C:of  which 
let  tiro  of  them,  Tvhkh  ttro 
fbeuer  be  taken*  be greater  then  the 
third y  that  is  let  the  lines  AjBAe 
greater  then  the  line  C,  and  the  lines 
Afijhen  the  line  B,and  the  lines  B, 

C*  then  the  line  \A .  It  is  required  of 

OwmtyMw » Jkmh 

lines  AfBfijomake  a  tmnglefTake 
a  right  line  baiting  an  appointed ende 
on  the  fide  £>,  and  being  infinite  on 
the  fide  E.  Andfbythe  $ ,  propopti* 
m)putVnto  the  line  A, an  e  quail  line :  *  '  * 

£) F,and put  ImtQ  the  line  Btan  equall  line  FG^and'vntoy  line  C3an  equall  line 

GH-  And  making  the  centre  Fyand  the  [pace  b  Ffiefcribe  (by  the  $  .peticio) 

a  circle  D  KJL.  Agayne  making  the  centre  G0and  the face  G  77,  deferibe  (by 

the fame )a  circle  H  I\L:and  let  the  point  of the  inter [eHion  of  the  fayd  circles 

be  K^andfby  the  fir fiptticion)dra'tr>  a  right  line  from  the  point  fitoy  point  Fy 

Cst*  an  other  from  the  point  fito  the  point  G,  Then  7 fay  3t  hat  of thre  right  lines 

equall  to  the  lines  AfBfijs  made  a  triangle  I\FG.  Forforafmuch  as  the  point  Dem»»ffr*t;o» 

F  is  the  centre  of  the  circled)  therefore  (by  the  definition)  the  lineF 

D  is  equall  to  the  line  F  If,  But  the  line  A  is  equall  to  the  line  F  D  Wherfore 

( by  tbefrfl  common  fentence )  the  line  F  Jfjs  equall  to  the  line  A.  Agayne  for* 

afmuch  as  the  point  G,is  the  centre  of the  circle  L  IfjA^therefcre(by  the  fame 

definition)  the  line  G  Kjs  equall  to  the  line  G  BBut  the  line  C  is  equall  to  the 

lineG  FL  therefore  (by  t  he  first  common  fentence)  the  line  K^G  is  equallto 

the  UneCfBut  the  line  FG  is  byfiuppofition  equal  to  the  line  B:  therefore  tbefe 

three  right  lines  GFfi K»,and  Kfifi  3are  equall  to  tbefe  three  right  lines  AfBy 

C.  Wherefore  of  three  right  lines  ,  that  is,  JfJF,F  G,  and  G  2^  ,  which  are 

equallto  the  thre  right  lines geuen that  is  to  A '  B.C,is  made  a  triangle  K  FG: 

M^as  required  to  be  done.  ?  ~  V 


CmjimStm „ 


An  other  conftru£tion,and  dcraonftration after  Flnffatcs* 

Suppofe  that  the  three  right  lines  be  Andyntq  fome  one  of  them3namely,  t,berc9mJ 

to  C,put  an  equall  line  D  £,and(by  the  fecond  propofitionjfrom  the  point  A,  draw  the  anf 

line  £  C/ , equall  to  the  lme^:  and  (by  the  fame)  vnto  the  point  D  put  the  lineDHe-  Irffiifrlt 

frt  hne/,Al;d  makLlnS the  centre  the  point  Ey  &  the  fpace  E  <?,defcribe  a  cir-  ^ 
cleFG  :  bice  wife  making  the  centre  the  point  .D,  andthe  fpace  D  H  deferibe  an  o- 

t  er  circle  which  circles  let  cutte  the  one  the  other  in  the  point  F.  And  draw 
-  -  I.ii;,  thefe 


%  t 


v  -  y<v 


thefe  lines  ©  F  and  E  F.  Then  I  faye 
thatf©££isa  triangle  defcribed  of 
§  right  lines  equal!  to  the  right  lines 
-^,£,C.For  forafmuch  as  the  line©// 
is  equall  to  the  right  line  ^4,  the  line 
©£.  (hall  alfobe  equall  to  the  fame 
light  line  For  that  the  lines  ©  H 
and  ©  £, are  drawen  fro  the  centre  to 
the  circumferenceJ.Likewife  foraf- 
much  as  E  ( 7 is  equall  to  E  F(by  the 
1 5«  definition)  and  the  right  line  B  is 
equall  to  the  fame  right  line  £  (7:ther 
fore  the  right  line  £  £  is  equall  to  the 
right  line/?  :  but  the  right  line  ©  £, 
was  put  to  be  equall  to  the  right  line 
C.Wherfore  of  three  right  lines  £  ©,©  F,  and  ££,Which  equa}  to three  right  lines’ 
geuen,  ts4,B,Cyis  defcribed  a  triangle:  which  was  required  to  be  done. 


» 


n:,-, 


.  5  h 


tnflances  in  this 
Frebleme. 


Fir  ft  inflame. 


& 


r  r  <-\  r  * 

*  •  V  •ci \  \  i .  :  .  ■  i  •  •  *  cv,  'A*'.  '-vJi  v  :  v  '  •  *n 

In  this  proposition  theaduerfary  paradueoturc  mil  caul  Ilshat  the  circles' 
fhall  not  cut  the  one  the  other  (which  thingfeuF/f  putteth  them  to  do)Butnosr 
ifthey  cutte  not  the  one  the  other, either  they  touch  the  one  the  ocher,or  they 
axe  diftaunte  the  one  from  the  other .  Fiiffcif  itbepoflibleiec  them tqoche the 
one  the  other:  as  in  the  figure  here  put  (the  conftruction  thereof  anfwercthto 
the  conftruaion  viEuclide). 

And  forafmuch  as  F  is  the  cen¬ 
tre  of  the  circle©  if.therfore  the  ; 
line  ©  £  is  equal  tothe  line  f’M 
And  forafmuch  as  the  point  C,is 
.  .  th  e  ce  ntte  o  f  the circle  HL,  ther- 

fore  theliiie  fi  G\  is  equall  to  the  j  \ 

line  G  M.  Wherefore  thefe  two  /  \ 

lines  ©  F,  and  G  H,  are  equall  to  ( 
one  line ,  namely ,  to  F  G .  But 
they  were  put  to  be  greater  then  9 
it :  for  the  lines  ©  F,  F  (?,  and  G  \ 

H,  were  put  to  be  eqiidll  to  the  \ 
lines  uery  two  of  which  ' 

are  fuppofed  to  be  greater  then 
the  thirds  ;  wherefore  they  are 
both  greater,  and  alio  equall,. 
tuami  infiance  which  is  impoffible.  Agayne  if  it 
be  poifibledct  the  circles  be  di- 
ftant  the  one  from  the  other,  as 
are  the  circles  DK&ndH  ©.And 
forafmuch  as  Fisthe  centre  of 
the  circle  ©  K,  therfore  the  line 
©  £  is  equal  to  the  line  F/V.And 
forafmuch  as  g  is  the  centre  of 
the  circle  L  H, .therefore  the  line 
HG  is  equall  to  the  line  G  Af: 
wherefore  the  whole  line  F  G  is 
greater  then  the  two  lines©  £, 
and  c j  Hy  (for  the  line  £  (/,exee- 
deth  the  lines  ©  £,  and  G  H,  by 
the  line  NM Jhut  it  was  fupjpo« 


ofEuclides  Elemntes* 


FoL]i. 

fed  that  the  linesDF  and  if  Care  greater  thm  the  Mne  FG :  as  alfo  it  was  fuppofed 
that  the  lines  A  and  C7wcre  greater  then  the  line  F(for  the line  D  Fisput  to  be  equal! 
to  the  line  A,  and  the  line  F  G  to  the  line  2?,  and  theline  H  G  to  th,e  line  C. )  Wherefore 
they  are  both  greater  and  alfo  equall:  which  is  impoffible.  Wherefore  the  circles  ney- 
ther  tooch  the  one  the  other,  nor  are  diftant  the  one  from  the  other.  Wherefore  of  ne- 
celfitie  they  cut  the  one  the  other:  which  was  required  to  be  proued. 


Ehe  p.  Tmbleme,  The  i^fPropofttwn. 

. v '  N\\  I  s  '  i 

Vpon  a  right  line geuenyand to  a point  in  it  geuen:  to  make  a 
reBiline  angle  equall  to  a  reBiline  angle  geuen* 

Vppofey  the  right  line  ge¬ 
ne  be  AB,  lety  point  in 

it  geuen  be  A.  And  let  alfo 
the  reBiline  angle  gene  be 
DCHJt  is  required  Vpon  the  right 
line  geuen  A  B,and  to  the  point  in  it 
geuen  A  Jo  make  a  reBiline  angle  e* 
quail  to  the  reBiline  angle  geuen  ID 
C  H.  Take  in  either  of  the  lines  CD 
and  CHa  point  at  all aduenturesjfr 
let  the  fame  he  Dand  E.And(by  the 
first  peticion)  dra'to  a  right  line  fro 
jp  to  E,And  ofthre  right  lines yA  F> 

F G  and  G  jiJfbich  let  be  equall  to 

the  three  right  lines  geuen  jhat  is, to  CDfD  E,and  EC,make(byy  proportion. 

goyng  hefore)a  triangle, and  let  the  fame  he  AEG;  fo  that  let  the  line  CD  be  e* 

quail  to  the  line  AF and  the  line  C  E  to  the  line  AG  }and  more  oner  the  lyne  D 

E  to  the  line  F  G.  And  fora fmuch  ds  thefe  t^o  lines  D  C  and  CEare  equall  to  Demonftr*tk*. 

thefe  tV>o  lines  FA  and  A  G,the  one  to  the  other, and  the  bafe  D  E  is  equall  to 

the  bafeF G'.ther fore  (by  the  S.  proportion)  the  angle  D  CE  is  equall  to  the 

angle  F AG  .W  her  fore  vpon  the  right  ImegeuenA  B tand  to  the  point  in  itge* 

uen  namely  A, is  made  a  reBiline  angle  F  A  Gjqual  to  the  reBiline  angle  geuen 

D  CH:  Tvhicb'SVas  required  to  he  done . 


.  CiitftrH&lot?, 


An  other conftrudionanddemonftration  after  Proclus, 

Suppofe  that  the  right  line  geuen  be  A  B:  &  let  the  point  in  it  geuen  be  A, 8c  let  the  Jn  other 
rediline  angle  geue  be  C  D  E, It  is  required  vpo  the  right  line  gene  A  B38c  to  the  point  jhuafLfZ de 
in  it  geue  A,to  make, a  redijine  angle  equal  to  the  rediline  angle  geue  CD  E.  Drawe  a  m0n(l ration af  s 
line  fro  C  to  E.  And  produce  the  line  A  B  on  either  fide  to  the  points  F  and  G.  And  vn-,  ter  Proclm. 

I.iiii.  to 


tt&  er  Tse- 
tswx-jtr# turn 
Pektariies, 


Tbefirmooke 


<  t  *  s  4  —  —  — ■•• » * —  —  ***»  .».*».*  %*  mw  €q  u  al  j  yjj* 

j°r  e-  LlnC  £.^PUC  c^e  ^ne  5(7  C(1  uallAnd  making  the  cetre  the  point  si, Sc  the  fpace  AF, 
4elcnbe  a  circle  K  F.  And  agayne  making  the  centre  the  point  B  and  the  fpace  B  G  de- 


fcribe  an  other  circle  G  L:  which  dial  of  necefiitie  cut  the  one  the  other,as  we  haue  be¬ 
fore  proued.Let  them  cut  the  one  the  other  in  the  pointes  M  &  N.And  draw  thefe  right 
lines  AN,  AM,  BN,  and  R  M.  And  forafmuch  as  F  A  is  equall  to  A  M:and  alfo  to  A  N 
(by  the  definition  of  a  circle)but  C  D  is  equall  to  F  A,whcrfore  the  lines  A  M  and  A  !M 
are  eche  equall  to  the  line  B  C.  Agayne  forafmuch5  a  sBG,is  equall  to  B'M,and  to  B  N, 
and  B  G  is  equall  to  C  E:  therfore  either  of  thefe  lines  B  M  and  B  N  is  equal!  to  the  line 
C  E  Bat  the  line  BA  is  equall  to  the  line  D  E.Wherfore  thefe  two  lines  B  A  &  A  M,arc 
equall  to  thefe  two  lines  D  E  and  D  C.the  one  to  the  other.and  the  bafeB  M  is  equal  to 
thebafeC  f.Wherforefby  the  8,propofition)the  angle  M  AB,is  equall  to  the  angle  at 
the  point  D.  And  by  the  fame  reafon  the  angle  N  A  B.is  equall  to  thefame  angle  at  the. 
point  O.Wherforevpon  the  right  line  geuen^i?,andtothe  point  in  it  geuen  ^4,is  de- 
feribed  a  reftilinc  angle  on  either  fide  of  the  line  ABi  namely, on  one  fide  the  re&iline 
angle  N  A  B,and  on  the  other  fide  the  re&iline  angle  M  A  B,  either  of  which  is  equall  to 
the  re&iline  angle  geueh  C BE:  which  was  required  to  be  done. 


An  other  conftru&ionalfo^and  detnonftration  after  Pelitarius.’ 

Suppofe  thar.the  right  line  geuen  be  AB:  and  let  the  point  in  it  geuenbe  C,  and 
letthere&ilineanglegeuen  be  DEFJtis  required  vpontheline  geuen  .//5,and  to  the 
point  in  it  geuen -C,to  deferibe  a  re&iiine  angle  equall  to  the  reitiline  angle  geuen  B  E 
JF.Prqduce  the  line  F  E  to  the  point  G :  and  from  the  point  E  ere£fc(by  the  1 1  .propofi- 
tion  )  vnto  the  line  GF  a  perpendictiler  line  E  H, which,  if  it  exa&ly  agree  with  the  lync 
iT'Z), then  was  the  angle  geue  a  right 
angle.Wherforeiffrom  thepointe  C 
you  erefte  a  perpendiculer  line  vnto 
the  line  ABythzt  fliall  be  done  which 

was  required  to  be  done  But  if  it  do  1G  K  t>  M 

notythenfrom  the  point//,  ereft  vn¬ 
to  the  line  HE,&  perpendiculer  lyne 
H'B  ,  whiche  being  produced  Hull 
(by  the  fifth  peticionjconcurre  with 
the  line  £  D  being  alfo  produced:  for 
the  angle  B  EH  is  Idfe  then  a  right 

angle(whenas6’£//isarightangje),  _ _ ,  _ _ , 

Wherefore  let  them  concurre  in  the  A  C  3  G  JS  p 

point  £>,audfois  made  the  triangle 
JE>  E  H.  After  the  fame  maner  fr5  the 

gsJica  C,em%  vnto  the  line  A  3  a  perpendiculer  line  C  K ;  which  let  be  equal!  to 

co  .ini  L 


the  perpendicular  line  EH( by  the  %  .pr6pofition):ahd  frcini  the  point  Ketcd  vntothfi 
line  K  C>a  perpendiculer  line  K Z^whiche  let  be  equall  to  the  perpendiculer  lyne  H  D* 
And  draw  a  line  from  C  to  E.Tben  I  fay  thatirhs  angle  L  C  F,is  equall  to  the  angle  ge¬ 
uen  DE  A.  lor  the  two  triangles  HE  CD  and  K G  L,are  (by  the  fourth  propofition)  e- 
qual,  and  cquilatcr  the  one  to  the  other;  and  the  two  angles  LCK  a  nd'DEH  are  equal* 
And  the  two  angIes|5C/C  and  EEH  are  equal,  for  either  of  them  is  a  right  angle.Wher- 
fote  (  by  the  i  kComffion  fentencejthe  whole atigleZ#  C  i?,iseqtiall  to  the  whole  angle  D 
E  F.  Which  was  required  to  be  done. 

And  if  the  perpendiculer  line  chaunce  to  fall  without  the  angl  egeuen4namely, if 
the  angle  geuen  be  an  acute  angle,  the  felfefame  manner  of  demonftration  will 
ferae:  but  onely  that  in  ftede  of  the  fecond  common  l'entencc,  muft  be  vfed  the 
2,common  fentencc* 

<  •  ■  ■  .  ’  . ; 

'  '  ........ 

Appollonius  putteth  another  conftru&ibn  Si  dcirionftratioh  of  this  propolitio: 
which  (though  the  demonftration  thereof  depende  of  proportions  put  in  the 
third booke,  y  et  for  that  the  conftrufhon  is  very  good  for  him  that  wil  redcly , 
and  mechanically,  without  demonftration,  defenbe  vpona  line  geuen,  and  toa 
point  init  geuen,  a  re&ilme  angle  equall  to  a  re&ilinc  angle  geuen)  I  thought 
not  amifle  here  to  place  it.  And  it  is  thus . 


Suppofe  that  the  refliline  angle  geuen  be  CD  E7  and  let  the  right  line  geuen  be 
ABS  and  let  the  point  in  it 
geuen  be  e^.  Take  in  the 
line  CU^a  point  at  all  ad¬ 
ventures  ,  which  let  be.F. 

And  making  the  cetre  the 
point  D,  and  the  (pace  D 
F,  deferibea  circle  A  (?,cut 
ting  the  line  in  the 
point  G,  and  draw  a  ryght 
line  from  F  to  G ,  Likewife 
from  the  line  AB,  cut  of 
a  line  equall  to  the  linei> 

FjWhich  let  be  AH.  And  Jq  jj 

making  the  cetre  the  point  ’  v 

A,  and  the  fpace  AH  Ac-  \ 

feribe  a  circle  H  K3  and  from  the  point  A/,fubtend  vnto  the  circumference  of  the  circle 
arightlyne  equal!  to  the  right  line  Fg ,  whicheletbee//FT;  and  drawe  a  right  lyne 
from  A  to  K.  Then  I  fay  thatthe  angle  HA  KM  equall  to  the  angle  CD  E,  The  proofs 
whereofl  now  omitte  for  that  it  dependeth  of  the  a  8  and  27  propofitions  of  the  third 
booke. 


rdft  etkiir  tm- 

Jlraf~ion  £<? 
menftrxtian  *f~ 

ttT 


But  now,as  I  fayd,  by  this  you  may  very  redily,  and  mechanically,  without  demon 
ftration,  vpon  a  line  geuen,  and  to  a  point  init  geuen  deferibe  a  re&iline  angle  equall  t6 
to  areftiline  angle  geuen.  For  ' 

in  the  rechlme  angle  geue,  you 
neede  onely  to  marke  the  two 
pointes,  where  the  circumfe¬ 
rence  of  the  circle  cutteth  the 
lines  contayning  the  angle  ge¬ 
uen:  as  the  points  Aand  G:  and 
likewife  to  marke  in  the  line  ge¬ 
uen  asm  AB,the  point  H,  8c  fo  a 

making  the  centre  the  point  ^according  to  the  fpace  AH  {which  is  put  to  be  equal  to 
FI)) deferibe  a  peece  of  a  circumference  on  that  fide  that  you  wil  haue  the  angle  to  he, 
asfor  example  the  circumference  H  AT,and  openingyour  compaffe  to  the  wideth  from 

K*i»  the 


■®#Mpidei  the 
firfi  tnuenter  of 
thii proportion. 


CtnUrtiftion, 


Dammft ration. 


the  point  F,  to  the  point  G,  fet  one  foote  thereof  fixed  iftthe  point  H,  and  marke  the* 
point  where  the  other  foote  cuttcth  the  fayde  circumference ,  which  point  let  be  Kt 
And  from  that  point  to  the  point  t^draw  a  right  line:  and  fo  (hall  you  hauedeferibed 
at  the  point  A,zn  angle  equal  tp  the  angle  at  the  point  I>. As  in  the  figures  in  the  end 
of  the  other  fide  put*  1 

O empties  was  the  firft  inuenter  of  this  proportion  as  witneffeth  Eudemi  us. 


1  iSTbeoreme  The  Z^fPropofition. 

fftwo  triangles  haue  two  fide s  of  the  one  e  quail  to  two fides  of 
the  other,  ech  to  his  correfpondent  fide, and  if  the  angle  cotai- 
ned  vnder  the  equall fides  of  the  One ,  be  greater  then  the  an¬ 
gle  contained  vnder  the  equall  fides  of  the  other :  the  hafe  alfo 
of the  fame, Jhalbe  greater  then  the  baje  of  the  other . 

V ppofe  that  there  he  Hbo  triangles  A  B  C,  and  1)  E  F} 
hauing  two.  fides  of  the  one ,  that  is,  A  B,  and  ji  C,  e* 
quail  to  two fides  of the  other,  that  is  jo  IDE,  and  D  F, 
ech  to  his  correfpondent fideithat  is, th fide  JtB,  to  the 
|  fide  V  E,and  the  fide  A  C  to  the  fide  DFx  and  fuppofe 
|  ’that  the  angle  $  AC  he  greater  then  the  angle  EVE 

2  f hen  I  fay  e  that  the  ha/e  BC,  is  greater  then  thebafe 
“  E  Ft  For  forafmuch  as  the  angle  B  AC  is  greater  then 


the  angle  ED  Fjnake  (by  the  23  -propofition)vp* 
on  the  right  line  DE,and  to  the  point  in  itgeueDy 
an  angle  ED  G  equall  to  the  angle  geuen  BAC. 

And  to  one  of  the fe  lines, that  is,  either  to  A  Cjr  D 
Eyput  an  equall  tine  D  G.  And  (by  the  firfi  peticio) 
draft  aright  line  from  the  point  G^td  the  point  E, 
and  an  other  from  the  point  F,  to  the  point  G.  And 
forafmuch  as  the  line  A  B  is  equall  to  the  line  V  E, 
and  the  line  AC  to  the  line  D  Gythe  one  toy  other, 
and  the  angle  BACis  (by  conflructm)  equall  to  .  7 

the  angle  ED  G,  therefore  (by  the q.propofition) the hafe  B C,  is  equal!  to 
she  hafe  E  G.Agayne for  as  much  as  the  line  D  G  is  equall  to  the  line  D  Father* 
( hy  the  5,  propofition)the  angle  DGFy  is  equall  to  the  angle  D  F  Gf/Vhere* 
fore  the  angle  DE G  is greater  then  the  angle  E  G  EjFFherefore  the  angle  E 
F  G  is  much  greater  then  the  angle  EGF And  forafmuch  as  EFG  is  a  trian # 
gle,  hauing  the  angle  E  F G greater  then  the  angle  E  G  F, and  (by  the  18. pro* 
pojition)  Vnder  the  greater  angle  is  fubtended  the  greater fide  ,  therefore  the 
fide  E  G  is  greater  then  the fide  E  FjBut  the fide  E  G  is  equall  to  the  fide  B  C; 
therefore  the  fide  B  C  is  greater  then  the  fide  E  F,  If  therefore  ttvo  triangles 
■  *•  'i  hmi  1 


ofEmMei  Elementeu 

haue  mo fides  of  throne  equal}  to  mo  fides  of  the  other ,  echeto  his  correfponfi 
dent  fide,  and  if the  angle  contayned  Vnder  the  e  quail fides  of  the  one  ,he grea¬ 
ter  then  the  angle  contayned  Vnder  the  equall fides  of  the  other :  the  bafealfo  of 
the fame  jhalbe greater  then  the  baje  of  the  other :  Which  Was  required  to  be 
proued . 

In  this  Theoreme  may  be three  cafes.  For  the  angle  S  JD  <7,  beingputequalltothe 
angled  A  C,  and  the  line  D  C^being  put  equall  to  the  line  AC}ind  a  line  beingdrawen 
from  £  to  G,  the  line  E  G  flulleither  fall  aboue'the  line  G  F ,  or  vpon  it,or  vnder  it.  £ «- 
^wdemonilrationferueth,  when  the  line  CZfaUethabbue  the  line  GFt  as  we  haue 
before  manifeltly  feene. 

But  if  it  fall  vpon  it,as  in  this  figure  here  piit. 

Then  forafmuch  as  the  two  lines  AB  and^C,are 
equal  to  the  two  lines  D  £andD  (/,thebne  to  the 
other,  and  they  contayne  equall  angles  by  con- 
ftrudion:  therefore  ('by  the 4.  propofitionj  the 
bafe  B  C,  is  equall  to  the  bafe  E  G :  but  the  bafe  £ 

<7,  is  greater  then  the  bafe£F  :  wherforealfo  the 
bafeFCyis  greater  then  the  bafe  £  F :  whichwas 
required  to  be  proued* 

But  now  let  the  line  £  (j,  fall  vnder  the  line  £ 

F, as  in  the  figure  here  put.  And  forafmuch  as  thefe 
two  lines  A  B ,  andACt  are  equall  to  thefe  two 
lines  D  E  and  D  (j,  the  one  to  the  other,  and  they 
contayne  equall  angles,  therefore  (by  the  4.  pro- 
pofition)the  bafe  B  Qis  equal  to  the  bafe  £  G.And 
forafmuch  as  within  the  triangle  DEG ,  the  two 
linnes  F  and  F  E> are  fet  vpon  the  fide  D  E:  ther- 
fore  f  by  the  2 1  .propofition)the  lines  JD  F  and  F  £ 
areieife  then  the  outward  lines *2)  G  and  G  Ex  but 
theline  D  G  is  equal  to  theline  D  F.  Wherfore  the 
line  G  E  is  greater  then  the  line  E  E.  But  G  E  is  e- 
quall  to  B  C. Wherefore  the  line  B  Cis  greater  the 
theline  £F,Which  was  required  to  be  proued. 


7  hree  ejfcs  i/i 
this  prop  aft  sen . 


The firft  cafe *. 
Second  cafe. 


Third  cafe. 


T>  (7,  which  arc  equal,  vnto  the  points 
K and  H :  and  draw  a  line  from  F  to  G: 
wherefore  (by  thefecond  part  of  the 
fifth  propofitidn)thc  anglesXF  G  and 
F  G  ft,  which  are  vnder  the  bafe  F  <7, 
are  equall  therefore  the  angle  E  FG  is 
greater  then  the  angle  FG  F.Wherfore 
(by  the  1 8  propofitionj the  fide£  <7  is 
greater  then  the  fide  £  JF;  but  the  bafe 
B  C  is  equal  vnto  the  bafe  E  G:  Where¬ 
fore  the  bafe  B  C,  is  greater  then  the 
bafe  £  F :  Which  was  required  to  be 
proued. 


produce  the  line  s  -Di7  and 
V  > 


£ 


An  other  de- 
monjlratien  of . 
the  third  caf. 


It  may  peraduenture  feme,  that  Euchde  fiiould  here  in  this  proportion  haue 
proued,  that  not  oncly  thebafes  ofthe  triangles  are  vnequall, but  alfo  that  the  as 
teas  of  the  fame  are  vnequallr  for  fo  in  the  fourth'pro  pofit  ion3  after  he  had  pro- 

Kii»  usd 


mty  EneliZe 
"hers  prouith 
*fot  :ht  areas  of 
-the  triangles  to 
:&<l  SsKSqttttlt, 


'lifter  thiff, 
2Jrtpajh/evJ*M 
ft  all fade  the 
campttrifsn  of 
■  triangles,  w  heft 
ijides  being 
equalljkefr  bit** 
Jes  and  angles 
•at  the  tcppe 
art  ineipnall. 


E>ems»f ration 
leading  to  ah 

abfterdttji 


ued  the  bate  to  be  equall,  he  protied  alfo  the  areas  to  be  equally  But  hereto  may 
be  answered,  that  in  equall  angles  and  bafes,and  vnequall  angles  and  hafes ,  the 
confideration  is  not  like*  For  the  angles  and  bafes  being  equall,  the  triangles  al¬ 
fo  fhall  ofnecefliticbe  equall,  but  the  angles  and  bafes  being  vnequall,  the  areas 
fliall  not  ofneceflitie  be  equall*  For  the  triangles  may  both  be  equall  and  vne- 
quail :  and  that  may  be  the  greater, whiche  bathe  the  greater  angle, and  thegred* 
ter  bafe,  and  it  may  alfo  be  the  lelfe.  And  for  that  caufe  Euclide  made  no  mend# 
on  ofthe  comparifon  ofthe  triangles*  Whereof  this  alfo  moughtbeacaiife,fot 
that  to  thedemonftration  thereof  are  required  ccrtayne  Proportions  concer- 
ning  parallel  lines,  which  vc  are  not  as  yet  come  vnto*  Howbeitafter  the  ^7, 
proportion  of  tins  booke  you  fhalfind  the  comparifon  of  thearcas  of  triangles* 
'which  haue  their  fides  equall,  and  their  bafes  andangles  at  the  toppe  vticquall* 

The  1 6.cTheoremv.  T'heifJPropofitiOtt* 

If  mo  triangles  haue  mo  fides  ofthe  otteequaU  tttmo fades 
ofthe  other ,  eche  to  bis  correfpondent fide, and  if  the  bafe  of 
the  one  be  greater  then  the  bafe  ofthe  other:  the  angle  alfo  of 
the  fame  cotayned  vnder  the  equall  right  lines  ffktitt  begrea^ 
ter  then  the  angle  ofthe  other* 

p  Vppofe  that  there  he  ftoo  triangles, A fB,Cy 
land  B)  E  FJiauing  vm>o  fides  0/  tb’onefbat 
Us,  A  B  ft  nd  AC  squall  toffyo  (ides  ofthe  0* 
therjhat  is  to  B)  Ey  and  B)  F,  ecb  to  his  correfpon* 
dent  fide, namely, the  fide  AB  to  the  fide  !DF,  and 
the fide  A  C  to  the  fyde  B)  F.  'But  let  (he  bafe  B  C 
be  greater  then  the  bafe  E  FfTbe  1  fay,  that  the  an* 
git  B  AC isgrea ter  then  the  an g  le  EB)F%  For  if 
nbt ,  then  is  'it  either  equall  Vn(6  it,  or  left  then  its 
But  the  angle  BAG  is  not  equall  to  the  angle. B  ID 
F: for  if  it  There  equall, tire  bafe  alfo  B  C Jhouldfbythe  ^propofitton)  be  equal 
to  the  bafe  E  F:  but  by  fuppofition  it  is  not.Wherfott  the  angle  B  AC  is  note* 
quail  to  the  angle  EDF.Neither  alfo  is  the angle fe  AC [left  then  the  angfiE 
£>  F:  for  then  fhould  the  bafe  BCbe  lefie  the  the  bafe  £  F(  by  the  former  pro* 
pofitioiifBut  by  fuppofition  it  is  notyFherfaef angle  BAGsmilefie 
anAe  E  B>  FJnd  it  is  already  protied, that  it  isnoietiuafomitMerfdrefia^ 
Ae  B  AC  is  greater  then  the  angle  EB)F.  If  therfiret^o  triangles  haue  iM 
fides  of  the  one  equall  to  nvo  fides  ofthe  other, eche  to  his  comfp*ndentfide,& 
if  the  bafe  ofthe  one  be  greater  then  the  bafe  ofthe  other ,  the  angle  alfd  ofthi 
fame  contaytied  Vnder  y  equal  right  lides jhdU  be  greater  the  the  angle  of theo . 

Tbit 


birstrs  eafis  i» 
this  dttmssflnt- 
tiers . 

fir&i*fi.'t 


oj-tsmttm  mtmmtes.  Fol.tf. 

This  propofitio  a  is  plaine  oppofiretb  the  el^ti  is  the  coucrfc  of  the  fdure  Hr/comZoZh 
and  twenty  which  went  befor^.^nd  it  is  proifed(as  commonly  ail  eonuetfes  are)  inJsreaij  <u- 
by  a  reafon  leading  to  an  ablufdi  cie*But  it  may  after  Menelaas  Alexand  rintrs  be  monftr*ted-  ^ 
demonilrared  directly  jafter  this  maner*  '  fUlll! 

Mentlaus  AUtr* 
andrin&s* 

Suppofe  that  there  be  two  triangles  iA B  C  8c  B 
£  F:haning  the  two  fides  A  B  and  Ca C  equal  to  the 
two  Tides  lD  E  and®  £,the  one  to  the  other:andle£ 
the  bafe  B  C  be  greater  then  the  bafe  E  /.Then  /  fay 
that  the  angle  at  the  point  A, is  greater  the  the  aft^ 
gle  at  the  point  ®. For  from  the  bafe  BCe  ut  of  (by 
the  thi  rde  ja  lineeqnallto  the  bale  £  F,  and  let  the 
fame  be  B  (?.And  vpon  the  line  GB  and  to  the  point 
B  putf  by  the  2  3,propofition)an  angle  equal  to  the 
angle  ©  E  Ft  which  let  be  G  B  H:  and  let  the  line  B 
H  be  equall  to  the  line  D  £.And  drawe  a  lync  from 
H  to  (band  produce  it  beyond  the  point  G:  whiche 
being  produced  lhall  fal  either  vppon  the  angle  Ay 
or  vpon  the  line  A  B, or  vpon  the  line  A  C ,  Firffc'  let 
it  fall  vpon  theangle  A.  And  forafmuch  as  thefe  two 
lines  B  G  and  B  H are  equall  to  thefe  two  lines  E  F 
and  E  D} the  one  to  the  other,and  they  contayne  e- 
quallanglesfbyconftru&ionjnamelyjtheangles  G 

BH and  D  E  F:  therforef  by  the  4.propofition)thei  bafe  G H is  equall  to  the  D  Ft  and 
the  angle®  H  G  to  the  angle  E  D  F.Agayne  forafmuch  as  the  line  B  His  equall  to  the 
jline  B  A(( or  the  line  A  Bis  fuppofed  to  be  equal  to  the  line  D  E,  vnto  which  line  the 
ineBH  isput  equal )  therforef  by  the  5  .propofition)the  anglcBFIA  is  equall  tothe 
angle  BAH :  wherfore  alfo  the  angle  E  D  F  is  equal  to  the  angle  B  A  //.But  the  angle  B 
A  C  is  greater  then  the  angle  BA  Hi  wherfore  alio  the  an  gle  BAG  is  greater  then  the 
angleEDF.  *  ^ 

But  now  let  it  fall  vpon  the  line  in  the 
point  K,  and  drawe  a  line  from  A  to  H.  And  for¬ 
afmuch  as  thefe  two  lines  B  G  andB  H  are  equall  to 
thefe  two  lines  E  F  and  E  D3the  one  to  theother,& 
they  containe  eq  ual  angles(by  conftrudion  j  liamc- 
lv,the angles GBH andD  E  F:  therfore (by  the  4* 
propofition)the  bafeGHis  equall  to  the  bafeD  F# 
and  the  angle  B  H  G  to  the  angle  EDF.Agayne  for¬ 
afmuch  as  in  the  triangle  B  A  H,the  fide  BA  is  equal 
to  the  fideB  H,therfore  (by  the  5  .propofition)the 
angle  B  A  FI  is  equal  to  the  angle  B  H  A.But  thean¬ 
gle  B  H  A  is  greater  then  the  angle  B  H  G; wherfore 
alfo  th  e  angle  BAH  is  greater  then  the  angle  BHG, 

Wherfore  the  angle  B  A  C  is  much  greater  then  the 
angle  B  H  G.But  it  is  proued  that  the  angle  BHG  is 
equall  to  the  angle  at  the  point  D  .  Wherefore  the 
dngleE.AC  is  greater  then  the  angle  at  thepbinte 
£):  Which  was  required  to  be  proued* 

But  now  fuppofe  that  the  line  H  G  beyng  produced  doo  fall  vppon  the  line  zAC, 
namely,in  the  point  K.And  agayne  draw  alfo  a  line  from  cA  to  //.And  forafmuch  as  B 
G  i  s  equal!  to  E  F,and  B  H,to  E  /^therefore  thefe  two  lines  B  G  and  BH  are  equall  to 
thefe  two  lines  E  F  and  E  D ,the  one  to  the  other,and  (by  conftrudion  Jthey  contayne 
equall  angles,namely,the  angles  GBH  and  FED, Wherfore  (by  the  fourth  propofitio ) 

K.iii.  the 


Thirdcstfe, 


je»  ttfter  de- 
■xns  nfiratisn  /tf‘ 
Ter  Hers  Mea 
eebaaicm. 


chc  bafejG  His  equal  tothebafc  *Z)  Ft  Sc  th’an 
gle  B  H  G  is  cquall  to  th’angle  E  D  E,  And  for¬ 
afmuch  as  GH is equall  toDi^andDF  ist- 
quall  to  AC:  therforeG  H  alfo  is  equal!  to  A 
C.  Wherfore  H  K  is  greater  then  A C,  where¬ 
fore  H  K  is  much  greater  then  A  K.  Wherfore 
(bytheiS.  propofition)  theangle  KAH  is 
greater  then  the  angle  K  H  A.  Agayne  foraf- 
much  a  sB  H  is  equahto  A  Bl(  for  B  H  is  pute- 
quall  to  D  £,  which  is  by  fuppofition  equal  to  B 
the  r  fore  (by  the  5  .propofition  ^  thean¬ 
gle  B  H  A  is  equall  to  the  angle  B  AH.  Wher- 
fore  the  whole  angle  BHK  is  leflethen  the 
whole  angle  BAK,  But  it  hath  bene  proued, 
that  the  angle  B  H  K  is  equall  to  the  angle  at 
•the  point  D,wherfore  the  angle  B  AC  is  grea¬ 
ter-then  tlie  angle  at  the  point  D,  which  was 
-re  q  u  ir  ed  to  b  e  p  r  o  ued. 

Hero  Mcchanicus  alfo  dcmonftratethican  other  way,andthat  by  4  dixcft 
•demonftranon. 


-a 


Suppofe  that  there  be  two  triangles  ABC 
and  DEE,  hauyngthetwofides-42?,and.// 

C,equallto  the  twofidesD  E,8cDF,  tlveone 
to  the  .other,  and  letthe  bale  BCy  be  greater 
then  the  bale  E  E. Then  I  fay,that  the  angle  at 
the  point -4,  is  greater  then  the  angle  at  the 
point  -D.Forforafmuch  a  sBCt  isgreater  the 
£F,produce£Ftothepoint£7,  and  put  the 
line  E  G,  equall  t“o  the  line  2 EC.  likewife  pro¬ 
duce  the  line  D  E  to  the  point  //,  and  put  the 
line  D  Hy  equall  to  the  line  D  E,.  Wherefore 
making  the  centre  the  point  D,and  the  /pace 
D  F,dcfcribe  a  circle,and  it  fliall  pa'fie  alfo  by 
the  paint  //.Let  the  fame  circle  be  E  K  //.And 
forafmuch  as  A  C  and  A  B  are  greater  the  BC 
(by  the  2  0.propofitio)&  the  lines  AB  &  A C, 
ar  equal  to  the  line  £H,8c  the  line  BC  is  equal 
to  the  line  E  (/.Therefore  theline  E  H  is  grea 
ter  then  theline  E  (J,  VVherefore  making  the 
centre  the  point  £  and  the  fpace  E  G  deferibe 
3tcirde,and  it  (hall  cut  the  line  E  //.  Let  the 
fame  circle  be  G  K:  and  from  the  common 
ife&ib  of  the  circles,  which  let  be  the’point  if, 
draw  rhefe  right  lines  K D  and  If  E.  And  for¬ 
afmuch  as  the  point  D  is  the  centre  of  the  cir¬ 
cle  H  K  F,  therefore^ by  the  1 5,  definition  JltheliijeD  K,is  equall  to  the  line  D  H, that 
is  vnto  the  fine  A  C.  Agayne  forafmuch  as  E  is  the  centre  of  the  circle  G  K,  therefore 
•theline  E  If  is  equal  to  theline  EG,thatis,to  the  line  B  C.  And  forafmuch  as  thefetwo 
lines  A  B  and  A  C,are  equall  to  thefe  two  lines  D  E  and  D  A", and  the  bale 2?  C is  equal 
to  the  bale  E  K(t or  E  Kis  equall  to  E  G  (by  the  1 5 .definition)  &  EG  is  put  to  be  equal 
to  B  C).Whereforef  by  the  4.propofitidn)the angle  B  A  Cis  equal  to  the  angle  E  D  K, 
But  the  angle  E  D  Ki s  greater  then  the  angle  ED  F :  wherefore  alfo  the  angle  B  A  C,if 
greater  then  the  angle  £  D  Ftwhich  was  required  to  be  proued. 


Tk  1 


ofEuclides  Elemenfes.  FoL]6 . 

The  17.  T htoreme.  (Thei6fPropoftim . 

.#2  j  _r j; “  ■' ’•  *  ''  1  -  . •.  s ;  ' •  • 

ffmo  triangles  haue  two  angles  of  the  one  equall  to  two  an¬ 
gles  of  the  other, ecb  to  his  correftondent  angle, and  haue  alfo 
one  fide  of the  one  equall to  one fide  of the  other ,  either  that 
fide  which  lie th  betwene  the  equal! angles,  or  that  which  is 
fukended  vnder  one  of  the  equallangles:  the  other jides  alfo 
of  the  onefalbe  equal!  to  the  other fides  of  the  other,  eche  to 
Us  correfpondent fide,  and  the  other  angle  of  the  one  jhalbe 
equal!  to  the  other  angle  of  the  other. 


Vppofe  that  there  he  t*too  triangles  A  B 
C,atid  D  E  E,hauing  Vtoo  angles  of  the 
one ,  that  is  .the  angles  A  BC,andB  C  A, 
e  quail  to  n>o  angles  of  the  other,  that  is, 
to  the  angles  ID  E  F,and  E  FD,ech  to  his  correfpo 
dent  angle, that  is, the  angle  AB  C,  to  the  angle  D 
EF,and  the  angle  B  C  A  to  the  angle  EFDyind  one 
fide  of  the  one  e  quail  to  one fide  of  $  other, firft  that 
fide  Ttshich  lieth  betlpene  the  equall  angles  ,  that  is, 
the  fide  BC, to  the  fide  EF.The  I  fay  that  the  other 
fides  alfo  of  the  one fhalbe  equall  to  the  other  fides  of  the  other, ecb  to  his  cor  re* 
f pendent  fide,thatis,the  fide  A  B,to  the fide  V  E,and  the  fide  A  C,to  the  fide 
DF,and  the  other  angle  of the  one, to  the  other  angle  of  the  other, that  is,  the 
angle  BA  C  to  the  angle  E  ID  FJFor  if the fide  A  B  be  not  equall  to  the  fide  T> 
Ejthe  one  of  them  is  greater ,  Let  the  fyde A  B  be  greater :  and  ( by  the  5  .propo 
fiiion)  Vnto  the  line  T>  E,put  an  equall  line  GB,and  drafts  a  right  line  from  the 
point  G,to  the  point  C.  ISLoSts  for  a  [much  as  thedine  G  B,  is  equall  to  the  line  D 
E,ahd  the  line  BC  to  the  line  E  F ^therefore  thefe  tstso  lines  G  B  and  B  C,  are 
equall  to  thefe  tSt>o  lines  ID  E  and  EFjhe  one  to  the  other, and  the  angle  GBC 
is  (by  fuppofition)  equall  to  the  angle  D  E  F.V therefore  (by  the  jp.propojy* 
tion )  the  bafe  G  Cis  equall  to  the  bafe  T>  F,  and  the  triangle  GCB  is  equall  to 
the  triangle  DUE, and  the  angles  remayningare  equall  to  the  angles  remay • 
ning  vnderTvhichare fubtended  equall fydes  .Wherefore  the  angle  GCB  is  o 
quail  to  the  angle  DFE.But  the  angle  D  EE  is  fuppofed  to  be  equall  to  the  an 
gle BC A.W here for e(by  the firfi common jentence)the angle  BCG'ts equal 
to  the  angle  B  CA,the  lefie  angle  to  the greater -.fvbich  is  impofiible .  Where- 
fore  the  line  A  B  is  not  Vne  quail  to  the  line  'DE.Wherefore  it  is  equall  And  the 
the  line  B  C  is  equall  to  the  line  E  F:now  therefore  there  are  tUso fydes  A  B  and 

Kjiq.  BC 


Dentonfirati.n 
leading  to  an 
abfurditk* 


iB  C  equal!  to  tV>o  fydes  t>  E  and  S  fi ,  the  one  to  tbeother,and  the  angle  AftC, 
if  equal!  to  the  angleD  EE .Wherefore(by  the  4-propofition)the  bafe  AC  is 
equal!  to  the  bafe  D  E^and  the  angle  remay ning  ft  AC  is  equal!  to  the  angle  re 
mayning  ED  F, 

Agaynefuppofe  that  the  fydes ful  tending  the  equaU  angles  be  equall  the 
one  to  the  other  Jet  the  fyde  I  fay  AD  be  equal!  to  the  fyde  D  E.  Then  agayne  l 
fiyjkat  the  other  fydes  of  the  one  are  equal!  to  the  other  fydes  o f the  other ,ech 
to  his  correfpondent  fyde, that  is  the  fyde  AC  to  the fyde  D  E,and  the  fyde  ft  C 
to  the  fyde  E  F :  and  moreouer  the  angle  remayning,  namely ,  ft  A  C,  is  equall 
to  the  angle  remayningy  that  is,  to  the  angle E  VF.  For  if  the  fyde  ft  C  he  not 
equall  to  the  fyde  E  F,the  one  of  them  is  greater:  let  the  fyde  ftCJfit  be  pofii* 
He ,  be  greater .  And  (by  the  third  proportion)  Vnto  the  line  EF,put  an 
equall  line  ft  H,and  dr  an?  e  a  right  line  from  the  point  J A  to  the  point  0,  And 
forafmuch  as  the  line  ft  Bis  equall  to  the  lineE  E ,  and  the  line  A  ft  to  the 
line  D  E, therefore  thefe  tn?oJydes  A ft  and  ft  0,  are  equall to  thefe  tn>o  fydes 
D  Eand  EF, the  one  to  the  other, and  they  contdine 
equa  11  angles, V Vi here  fore  (by  the  A.propofitmi)  the 
bafe  A  His  equall  to  the  bafe  D  E,and  the  triangle 
A  ft  H,is  equall  to  the  triangle  D  E  F^and  the  an*  c 

gtes  remayning  are  equall  todhe  angles  remayning, 

Vnder'tobtcb  ar  fubteded  equalfydes.VVherfore  the 
angle  ft  HA is  equall  to  the  angle  FED.  ftut  the 
angle  E  ED  is  equall  to  the  angle  ftCJ \  Where * 
fore  the  angle  ft  HA  is  equal  to  the  angle  ft  C  A \ 

Wherefore  the  outward  angle  of y  triangle  A  HC, 
namely, the  angle  ft  0  Ads  equall  to  the  inlvard  and  oppofite  angle  yiamely , to 
the  angle  HCAft»hich(by  the  \6  proportion)  is  impofible.Wherfore  the  fyde 
E  Fis  not'  Unequal!  to  the  fyde  ft  Cohere  fore  it  is  equall.  And  the fyde  A  ft  is 
equall  toy  fyde  D  Ei'toherefor  e  thefe  t'Hoo  fydes  Aft  andftC,are  equall  to  thefe 
two  fydes  DE  and.E  F,the  one  to  the  other,  and  they  contayne  equall  angles : 
W  her  fore  (by  the  ^propofitm  )the  bafe  A  C  is  equall  to  the  bafe  D  E:and  the 
triangle  A  ft  C,is  equall  to  the  triangle  D  EF,and  the  angle  remay  ning,name* 
fy,the  angle  ft  AC  is  equall  to  the  angle  remayning,  that  is, to  the  angle  EDF. 
If  there  fore  two  triangles  haue  Wo  angles  of  the  one  equall  to  tH>o  angles  of  the 
other,  ech  to  his  correfpondent  angle, and  haue  alfo  one  fyde  of the  one  equall  to 
qne. fyde  of  the  other, either  thatfydefohich  lietb  betToene  the  equall  angles,  or 
that  tohich  is  fub  tended  Vnder  one  of  the  equall  angles :  the  other  fydes  alfo  of 
the  one fhalbe  equall  to  the  other  fydes  of  the  other,  eche  to  his  correfpondent 
fide, and  the  other  angle  of the  one  fhalbe  equall  to  the  other  angle  of  the  other : 
Ifibicb  laces  required  to  be proued* 

Whereas  in  this  propofition  it  is  fay  de,  that  triangles  are  equall ,  which 
featuring  two  angles  of  the  one  equall  to  two  angles  ofthe  other, the  one  to  the  o- 

thcr, 


of Euclides  Elements .  FoLff. 

ther,  haue  alfo  one  fide  ofthc  one  cquall  to  one  fide  of  the  other,either  that  fide 
which  lieth  betwene  the  equal!  angles,  or  that  fide  which  fubtendeth  one  of  the 
equall  angles :th  is  is  to  be  noted  that  without  that  caution  touching  the  cquall 
fiie,the  propofition  fiiall  not  alway  e$  be  true.  As  for  example. 

Suppofe  that  there  be  a  rectangle  triangle  A  B  C.whofe  right  angle  let  be  at  the 
point  B,Sc  let  the  fide  B  C  be  greater  the  the  fide  B  ^4:and  produce  the  line  A  2?,f ro  the 
point  'B  to  the  point  D.And  vpo  the  right  line 
B  C  &  to  the  point  in  it  C,  make  vnto  the  angle 
B  AC  an  equal  angle(by  the  2  3 .  propofition), 
which  let  be  BCD ,8c  let  the  lines  BD  Sc  CD, be 
ing  produced  cocurrein  the  point  D .Now  thS 
there  are  two  triangles  A  B  C,and  BCD,w hich 
haue  two  angles  of  the  one  equall  to  two  an¬ 
gles  of  the  other,the  one  to  the  other,namely, 
the  angle  *ABC  to  the  angle  DBC  (for  they 
are  both  right  angles ),  Sc  the  angle  B  A  C,  to 
the  angle  BCD( by  conftru&ionjand  haue  al¬ 
fo  one  fide  of  the  one  equall  to  one  fide  of  the  other,  namely, the  fide  B  C,  which  is  co¬ 
ition  to  them  both.  And  yet  notwithftanding  the  triangles  are  not  equall  :for  the  tri¬ 
angle  B  DC,is  greater  then  the  triangle  AB  C.Forvpon the  right  line  BC,  and  to  the 
point  in  it  C,defcribe  an  angle  equall  to  the  angled  CB:  which  let  be  FCB( by  the  2  3 . 
propofition ).And  forafmuch  as  the  fide  B  C  was  fuppofed  to  be  greater  then  the  fide, 

AB ,  therefore  (by  the  1  S.propofition)  the  angle  B  AC  is  greater  then  the  angle  BC 
^,whereforealfotheangle5CDisgreaterthentheangle^CJP.  Wherefore  the  tri¬ 
angle  BCDis  greater  then  the  triangle  B  £f.  Agayne  forafmuch  as  there  are  two  tri¬ 
angles  A  B  £and  B  Cishauing  two  angles  of  the  one  equal  to  two  angles  of  the  other*, 
the  one  to  the  other,namely,the  angle  ABC  to  the  angle  FBCffor  they  are  both  right 
angles)  and  the  angled  CB  to  the  angle  FCB(by  conftru&ion),and  one  fide  of  the  one 
is  equall  to  one  fide  of  the  other,  namely,that  fide  which  lieth  betwene  the  equall  an- 
gles.thatis.the  fide  B  C  which  is  common  to  both  triangles.  Wherefore  (by  this  pro¬ 
pofition)  the  triangles  A  B  Cand  F  B  Care  equal. But  the  triangle  DBC  is  greater  the 
the  triangle  F2?C.  Wherefore  alfo  the  triangle  D  B  fis  greater  then  the  triangle  A  B 
C. Wherefore  the  triangles  ABC  and  D  B  C,  are  not  cquall.-notwithftanding  they  haue 
two  angles  of  the  one  equall  to  two  angles  of  the  other,the  one  to  the  other ,  and  one 
fide  of  the  one  equall  to  one  fide  of  the  other. 

The  reafon  wherof  is, for  that  the  equal  fide  in  one  triangle,  fubtedeth  one  of 
the  equall  angles, and  in  the  other  lieth  betwene  the  equal  angles. So  that  you  fee 
that  it  is  ol  necclfitie  that  the  equall  fide  do  in  both  triangles, cither  fubtend  one 
of  the  equall  anglcs,or  lie  betwene  the  equall  angles. 

Of  this  propofition  was  Thales  Milefius  theinuentor,  as  witncffethEude- 
inus  in  his  booke  of  Geometricall  enarrations,  Thales 

the  snuentcr  of 
this  fro fojstitss. 

clhe  i&FTheoreme.  Theij-Tropo/ttion* 

If  a  right  line  falling  vpon  two  right  lines  Jo  make  the  alter ~ 
nate  angles  equall  the  one  to  the  other :  thofe  two  right  lines 
are parallels  the  one  to  the  other . 


c  . 


ThefrflTdooke 

\  V ppofe  that  the  right  line  E  F falling  Vppon  thefe  Mo  right  lines  A  B 
\and  C  D^o  wake  the  alternate  angles  ^namely, the  angles  A  EF<&  E 
*F  V  equall  the  one  to  the  other  Ah  enl fay  that  A  B  is  a  parallel  line  to 
C  D.  For  if  not 3  then  thefe  lines  produced fhall 

mete  together  ^either  on  the fide  of  B  and  (Dyor  on 
23  njlrattan  the  fyde  of  A  ip  C.Let  them  be  produced  therfore , 
let  ri)em  mete  lflt  be  pofiible  on  the  fyde  of  B 
andVjn  the  point  G.  VF her  fore  in  the  triangle 
G  E  Ffthe  ouMar dangle  A  E  Fis  equal  to  the  m - 
trard and  oppofite  angle  E  F Gftohich  (by  the  f6- 
propo(ition}isimpofiible4  Wberfore  the  lines  AB 
and  C(D  beyng produced  on  the  fide  ofB  and 

jhallnot  meeteJn  like  forte  alfo  may  it  be  prouedthat  they  fhall  not  mete  on  the 
fyde  of  AandC ,  But  lines  Tbhiche  being  produced  on  no  fydemeete  together yare 
parrallell  lines  (by  the  lafi  definition:)wherforeA  B  is  a  parrallel  line  toC  T)Jf 
therfore  a  right  line  fallings pon  Mo  right  lines  ,do  make  the  alternate  angles 
equall  the  one  to  the  other:  thofe  Mo  right  lines  are parrallels  the  one  to  the  o* 
then  Ttthkh  'to&s  required  to  be  demonstrated . 


Tbit  mrde  al¬ 
ternate  9 ft  A  in 
tkmcn fenfet*. 

Hots  it  is  hdgtt 
in  this  plats. 
Whu  h  singles 
are  csUedalters 
fusts. 


This  worde  alternate  is  ofEuclide  in  diuers  places  diuerfiy  taken:  fomtimes 
forakind  offituation  in  place,andfomtimefbran  order  in  proportion,  in  which 
fignification  he  vfeth  it  in  the  v.booke,and  in  his  bokes  of  numbers.  And  in  the 
firftfignification  he  vfeth  it  here  in  this  place,  and  generally  in  all  hys  other 
bokes  ,hauing  to  do  with  lines  Sc  figures* And  thofe  two  angles  hecallech  alter¬ 
nate,  which  beyng  both  contayned  within  two  parallel  or  equidiftant  lynes  arc 
neither  angles  in  order,  nor  are  on  the  one  and  felfe  fame  fide,  but  are  feperated 
the  one  from  the  other  by  the  line  which  falleth  on  the  two  lines:  the  one  angle 
beyngaboue,and  the  other  beneath. 


T he  iy.Theoreme.  The  z8.  Tropoftion. 

ffa  right  line  fatting  ypon  two  right  lines ,ma^e  the  outward 
angle  equall  to  the  inward  and  oppofite  angle  on  one  and  the 
fame  fyde, or  the  inwar de  angles  on  one  and  the  fame  fyde ,  e- 
quail  to  two  right  angles ithoje  two  right  lines ) hall  be paraU 
lets  the  one  to  the  other • 


Vppofethat  the  right  line  EF^  fallyng  Vppon  thefe  Mo  right  lines 
A  B  andCfDfio  make  the  outward  angle  EGB  equall  to  the  inward 
and  oppofite  angle  G  H  T>,or  do  make  the  inward  angles  on  one  and 

the 


of  Euelides  Element es. 


FoLfi. 


the  fme  jidejbat  is,  the  angles  BGH  andGHD 
equall  to  tibo  right  anglesXhen  I  fay  that  the  lyne  A 
IB  is  a  parallel  line  to  the  lyne  C  DEorforafmuchas 
the  angle  E  G  Bis(byfuppofition)equall  to.  the  an* 
gle  G  HD^and  the  angle  E  G  B  is(by  the  1 5  .pro* 
pofition)equdll to  the  angle  jiGH:  therfore  the  an* 
gle  AJj...  ft  he quail  to  the  angle  G  HD :  and  they 
are  alternate  angles.VVherj or e( by  the  zy. pro  pop* 
tUn)  AB  is  a  parallel  line  to  C  ID. 

Agaynefdrafmuch  as  the  angles  BGH  and  G  HD  are  (by  fuppo/ition)e» 
quail  ionvo  right  angles  &  ( by  the  i^propofition )tbe  angles  J.  G  H  and  BG 
H,ar  ealfo  equall  to  tVo  right  angles,  therefore  the  angles  AG  Hand'B  G 
H}  are e  quail  to  the  angles  BGH  and  G  HD:  take  avay  the  angle  BGH 
vhich  is  common  to  them  both  Wherfore  the  angle  nmainyng,namety,AGH 
is  equall  to  the  angle  remay  ning,namely, to  G  H  D.And  they  are  alternate  an* 
gles.  VVherfore(by  the  former  proportion)  A  B  isaparallell  line  to  CD ,  If 
therfore  a  right  line  fallyng  Vpon  tvo  right  lines, do  make  the  outward  angle  e* 
quail  to  the  inward  and  opposite  angle  on  one  and  the  fame  fide, or  the  invar  de 
angles  on  one  and  the  fame  (ideyequall  to  tVo  right  angles, thofe  tVo  right  lines 
Jhall  be  parallels  the  one  to  the  otherivhich  Vas  required  to  be  proued , 

Ptolomeus  dcmonftrateth  the  fecond  part  of  this  propofition,  namely,  that 
the  two  inward  angles  on  one  and  the  fame  fide  being cqual^the  right  lines  arc 
parellels,afterthis  manner* 

Suppofe  that  there  be  two  right  lines  A B  and  C  T), and  let  a  certaync  right  line  E 
FCj  H  cuttethemin  fuche  forte,  that 
it  make  the  angles  B  F  g  and  FGD  e- 
quail  to  two  right  angles.  Then  I  fay, 
that  thele  right  lines  zAB  and  CD  are 
parallel  lines,that  is, they  fhall  not  con- 
curre.For  if  it  be  poffibledet  the  lines  B 
F  and  G  D  being  produced  concurrc  in 
thepointeK.  Noweforafmucheas  the 
right  line  E  F  itandeth  vppon  the  right 
line  A 2?,therfore(by  the  1  ^.proporti¬ 
on  Jit  maketh  the  angles  zA F  £,  and  B 
FE  equall  to  t  wo  right  angles :  Iikewife  forafmuch  as  the  line  E  g  ftandeth  vpo  the  line 
C  Z^therforef  by  the  fame  propofition  Jit  maketh  the  angles  CG  F  and  D  g  F  equall  to 
two  right  angles.  Wherfore  the  foure  angles  B  FE,  zAF  E,  CG  F,  and ‘Z>  C7  Fare  equal 
to  foure  right  angles :  of  which  the  two  angles  B  F  G  and  FGD  are(by  fuppofition)  e- 
quall  to  two  right  angles,wherfore  the  angles  remaining,namely,  zAV  G  and  C(?F  are 
alfo  equall  to  two  right  angles  .if  therfore  the  right  lines  F  B  and  G  D  being  produced 
(the  inward  angles  being  equall  to  two  right  angles  Jdo  concurre,  then  fhall  the  lynes 
FA3.n6.GC  being  produced  concurre.  For  the  angles  AY  G  and  CG  F  are  equall  to 
two  right  angles.For  either  the  right  linesfhall  concurre  on  either  fide,  or  els  on  nei¬ 
ther  fide.For  that  on  either  fide  the  angles  are  equall  to  two  right  angles.  Wherefore 
let  the  right  lines  FA  and  GC  concurre  in  the  point  L.Wherefore  the  two  right  lines 
LAFKandLCGK  do  comprehends  ipace,  which  (by  the  6.  petition) is impofiu 

L.ij,  ble. 


S 


DmtnftrnMis 


An  $ther  cUm$- 
Jlrnthn  of  the 
fccondptsrt  af 
this  prtfffitien 
after  i'tekmex* 


ThefirftBooke 

He.Wherfore  itis  not  poflible  that  the  inward  angles  being  equal  to  two  right  angled 
the  right  lines  fhould  concurrc.  Wherefore  they  are  parallels :  which  was  required  to 
beproucd.  '  c  ^  5  •  n 


ittmznfifAUon 
Seeding  ft  <f* 

fjrftgart. 


The  lo.Theoreme.  The  zyfPropofition. 

...  '  ■  ■ 

fright  line  line falling  rppon  two  parallel  right  lines :  ma- 
hpththe  alternate  angles  equall  the  one  to  the  other :  andal- 
fo  the  oumarde  angle  equdtl  to  the  inwarde  andoppojite  an- 
glean  one  and  thejamefideiand  moreouerthe  inwarde  an¬ 
gles  on  one  and  the  fame  fide  equall  to  two  right  angles. 


Vppofe  tlxttppon  theft  parallel  lines  A  Band  C  IDd&fal 
the  right  line  E  F.  Then  I  fay  that  the  alternate  an* 
gles  ibbkk  it  maketh,  namely  r  the  angles  A  G  H  and 
G  H  Bf^are  equad  the  one 
to  the  other. andy  the  out* 
li?ard  angle  EGBis  equal 
\to  the  in'toarde  and  oppo* 
fite  angle  on  the  fame: fide 9 
namely yto)  angle  G  H  B:andj  the  inward  an¬ 
gles  on  one  and  the  felfe  fame  fide ythat  is^the  an -  c 
ales BG  Band G  HT>, are  equad  to  two  right 
Angles. For  if  the  angle  AGHbe  not  equal  to  the 
angle  G  H  ID, the  one  of  them  is greater. Let  the  angle  AGH  he  greater.  And 
for af much  as  the  angle  AG  His  greater  then  the  angle  GH  Vrput  the  angle 
B  G  Hcommo  to  theboth.Wherforey  angles  A  GHand  BGH,  aregreater 
the y  angles  BG  H&GB iD.  But by  ey  i$ .  propofitio)  angles  AGHzsrB 
GHare  e  quad  to  ttfo  right  angles, "toher fore  y  angles  BGH  er  GHD  arelefie 
the  two  right  angles. But  (hy)  $. petition )  ifvpotWo  right  lines  do  fall  a  right 
line  yaking)  inward  angles  on  one  and y  fame  fide, le fie  the  Ctoo  right  angles, 
thofe  right  lines  being  inftmtly  produced  muft  ncedesaty  length  meete  on  the. 
fide  Ipherin  are  the  angles  kffe  the  tSto  right  angks^VVherfore  the  right  lines 
A  B  and  C ID  being  infinitely  produced  'toillat)  length  meete. But  they  cannot 
meete, becau ft  they  are  paradels(hyfiuppofition):^berfore  the  angle  A  G  Bit 
not  -vnequaU  to  the  angle  GHDi'tokerfore  it  is  equad. 

And  the  angle  A  G  His(by  the  i<;.propofition)equall  to  the  angle  EG  % 
VVherfore  (by  the  fir  ft  common fentence)  the  angle  E  ’Q  Bis  equad  to  the  an* 

^  But  the  angle  B  G  H common  to  them  botkwherfore  the  angles  EG  B 

and  B  GH,are  equall  to  the  angles  BG  Hand  G  HV.But  the  angles  RGB 

and 


ofEucluIes  Elements*  FoL]p. 

md  <BGH  are  (by  the  i^propofition)eqmM  to  ftoo  ri*h  t  angles ,  VJ? he  re  fore 
the  angles  BGB  and  G  HD  are  alfe  equattmtm-  right  angles.  If  a  lyne 
therfore  do  fall  Vpo  two  parallel  right  lines:kmd^tly  th£naltermte^arigles  equal 
the  one  to  the  other.and alfo  the  outward  itngieftq-udltd  the  inipard  and  oppoo 
fhe  angle' on  one  and  the  fame  fide\ and  momm  et  the  inward  angles  on  one  and 
the  fame  fyde  equallto  tSho  right  angles:  yahiche  'Was  required  to  he  demon* 
Jirated,  . 

This  proportion  is  the conuerfe  ofthe  'two  proportions  next  going  before. 
Fo^that  which  in  cither  of  thernjs  the  thing  fought,or  c5cluron}is  in  this  the 
tin  ng  geucnjor  fupportiotfi And  contrati wile  the  thiri'gcs  which  in  them  were 
geuen  orfupportions3areinthis  proued,ahdafecidhciiirons4 

.4  ■.  '  '•  :v\  ‘  ''A  f\  v V.'Vv  V>  : 

’Telit arins after  this  proportion  addeth  this  witty  conchifioiv* 

■  .  . ; '  '  .  .  .  -  ... 

I- '  F  ■  V  '  •’ 

If  two  right  lines  tybich  cut. Wo  parallellipes  filo  be  Went  thefajde  parallel  lines  conairrs  in  a 
point fdndmahe  the  alternate  angles  equally  or  the  onward  angle  equall  to  the  irWardand  oppofite 
angle  on  the fame  fide  ,cr finally  the  tveo  inward  angles  on  one  and  the  felfe  fame  fide,  equall  to  Wo  right 
angles  uhofe  Wo  right  Imes  are :  draft  endiretlly  and  make  one  right  line. 

Suppofe  that  there  be  two  right  lines  ex/  B  and  Gif, which  let  cut  two  parallel  lines 
JD  £  and  F  G:  and  let  ^  B  cut  the  line  D  E  in  the  pOintAAand  let  C  B  cut  the  line  F  G  in 
the  point  K:8c  let  the  lines  A  B  ScC  B,c oncurre  betwene  the  two  parallel  lines  DE@z 
F  G  in  the  point  B  :  and  let  the  angle  D  HBbt  s- 
qualto  the  angle  B  K  G :  or  let  the  angle  A  HD  be 
equall  to  the  angle  B^K-F:  orfinally  let  the  angles 
B  H  D  and  B  KF  be  equal  to  two  right  angles.Thc 
I  fay  that  the  two  lines  ABandBC  are  drawen  di 
re£lly,.and  do  make  one  right  line.  For  if  they  be 
not,then  produce  AB  vnti!  it  cut  FG  in  thepoint 
Ii.anH  let  A  L  he  one  righ  t  line, and  fo  ihal  be  made 
the  triangle  B  L  K.  Now  then  (  by  the  firft  part  of 
this  29.pfopofition)the  angle  DHB  dial  be  equal 
to  the  alternate  angle  (]  L  B:  but(by  fuppofitionj 
the  angle  D  H  Bis  equall  to  the  angle  A  KG.  Wherefore  the  angle  B  L  G is  equall  to 
the  angle  B  KL}  namely,  the  outward  angle  to  the  inwarde  and  oppofite  angle:which 
(by  the  i  <5.propofition Jis  impolfible. 

Moreouer  ( by  the  fecodpart  of  this  29.  propofitio  Jthe  angle  A-H-D  lhalbe  equal 
to  the  angle  B  L  AT,  namely,  the  outward  angle  to  the  inward  and  oppofite  angle  on 
one  and  the  fame  fide.  But  the  fame  angle  AH  D  is  fuppofed  to  be  equal!  to  the  angle 
B  K  Fiwherefore  the  angle B  K  Fis  equall  to  theangle B  L  A'. Which  (by  the  felfe  fame 
1 6.  propofition)  is  impolfible.  *  ,  -  .  (  . 

.  Laftly  forafmuchas  the  angles  B  HT)  and  B  KF  are  fuppofed  to  beeqtiall  to  twfO 
right  angles,  &  the  angles  B  H  D  &BLKatealfo  by  thelaftpart  ofthis  appropofiti- 
on  equal  to  two  right  angles,therefore  the  angle B  K  F  lhalbe  equal  to  the  angle-B  LK; 

which  agayne  by  the  felfe  fame  i£-propofition  is  impolfible, 

IhezifEheoreme  The  ^.Tropofition* 

"Might  lines  which  are  parallels  to  one  and  the  felfe  fame 
right  lineiare  alfo parrallel  lines  the  one  to  the  other . 

L.Hj.  Suppofe 


7  his prop  oft  toss 
is  the  conuerfe 
ofthe  two  for¬ 
mer  propofttisi. 


An  addition  of 
Pelitarim. 


Demonftratim.  . 
leading  tp  an 
alfitrditie t 
Fir  ft  part. 


Second  pare. 


Third  part? 


ThefirftBooke 

Vppofetbat  thefe  right  lines 
f,  A  B  and CD, be  parallel  lines 
to  theright  line  EF.  Then  I 
Jay, that  the  line  A  B  is  a  parallel  line 
foC<D.Let  there  fatlDpon  the/e  the 
lines  a  right  line  G  HKf  Andforaf • 
much  as  the  right  line  G HI f  falleth 
yppon  thefe  parallel  right  lines  AB 
and  EF,therfore(by  the  prop  oft  ion 
~^VJ:uvV<\a  going  before) the  angle  A  G  H  is  e* 
quail  to  the  angle  G HR. Agayne  for* 
ajmuch  as  the  right  line  G  If  falleth 
yppon  thefe  parallel!  right  lines  E  F 
and  C  D ^therefore  (by  the  fame)  the 


A  ..  . 

a/ 

B 

E 

/h 

jr 

T.  •>  .  -* 

i  / 

r-£  -i  >•  ",  * 

C  ■  - 

-  .  '  ]  f'r‘  ■ 

/& 

A 

•1;  1  T~  ' •  • 

Anciheretefem- 
sh&e  Prsile/zutf  1 


angle  GHF'u  equal! to  the  angle  G  JfJD.lS{pTi>  thenit  proued  that  the  angle  A 
G  H  is  equal!  toy  angle  G  HE,  andy  the  angle  G  KJD  is  e quail  to  the  angle 
G  HF.V E her  fore  the  angle  jiG  Kjs  e  quail  to  the  angle  G  KJD,  And  they 
are  alternate  anglesiwherfore  AB  is  a  parallelline  to  C  DRight  lines  therfore 
ft  hid?  are  parallels  to  one  and  the  felfefame  right  line, are  alJo  paraM  lines  the 
me  to  the  other:  Trbicb  y?as  required  to  be  proued . 

EuciideirathedemonfkationofthisprQpofmQn,  fetteth  the  two  parallel 
Isneswhich  are  compared  to  one,  in  the  extremes ,  and  theparrallcl  towhomc 
they  are  compared,he  piacetli  in  the  middle ,  for  the  eaficr  demonftration .  Ic 
may  alfo  be  proued  euen  by  a  principle  onely.  For  if  they  fhouldc  conairrc  oa 
any  oncfide,thcy  {houldconcurrealfo  with  the  middle  iinc^andfojfliould  they 
not  be  parallels  vnto  it, which  yet  they  are  fuppofed  to  be* 

But  if  you  will  altertheir  pofition  and  placing.andfet  that  line  to  which  you 
will  copare  the  other  two  lines  ,aboue,or  beneath:  you  may  vie  the  famedemon 
fixation  which  you  had  before.  As  for  example. 

Suppofe  that  the  fines  A B  and  CD  be 
parallels  to  the  fine  £  F :  and  let  both  the 
lines  A  B  and  CD ,  be  abouc ,  and  let  the 
line  EF  be  beneath,and  not  in  the  middeft. 

Vpon  which  -let  the  right  fine  GH  Ki all. 

And  forafmuch  as  either  of  the  angles  KH 
DandAfC-Sisequall  to  the  angle  H  KE9 
(for  they  ar  alternate  angles  Jtherforc  they 
are  (by  the  firft  common  feutence  Jcquall 
'the  one  to  the  other.Whereforefby  the  28 
propofition) the  right  lines  AB  and  £F» 
are  parallels. 

But  here  ifa  man  will  obieft  that  the  lines  E  K  and  K  F,  are  parallels  vnto 
she  line  C  D,and  therefore  are  parallels  the  one  to  theother.  V  Ve  will  anfwere 
that  die  lines  E  K  and  K  Fare  partes  of  one  parallel  line,and  are  not  two  parallel 

lines. 


A 

A 

B 

c 

V 

E 

A 

r 

FoL\  o. 


liaes.For  parallel  lines  a  r  vnderffandedtobe  produced  infirmly  But  E  K  being 
producedifalleth  vpou  K  F*Wherefore  it  is  one  and  the  felfe  fame  with  it,  and 
not  an  other,  wherefore  all  the  partes  of  a  parrallelline  are  parallel  s ,  both  to  the 
right  line  vnto  which  the  whole  parallel  line  is  a  paralleled  alfo  to  al  the  parts 
of  the  fame  right  line*  As  thelineEK  is  a  parallel  vnto  HD,and  the  lineK  Eto 
the  line  C  H*  For  if  they  be  produced  infinitly  ,they  will  neuer  conciirre, 

Howbeit  there  are  fome  which  like  not,  thattwo  diftindt  parellel  lines, 
fhouldbe  taken  and  counted  for  one  parallel  line:  for  that  the  continuall  quan¬ 
tity  ,namely, the  line  is  cutafonder,andcefTeth  to  be  one*  Wherefore  they  fay, 
that  there  ought  to  be  two  diftind  parallel  lines  compared  to  one.  And  therfore 
they  adde  to  the  proportion  acorredion,  in  this  man er.  Two  lines  being  parallels  to 
one  line ;  are  either  parallels  the  one  the  other ,or  els  the  one  is  fet  dirtRly  again  fie  the  other 3fo  that  if 
they  be  produced  they  fliouldmake  one  right  line.  As  for  example * 


H 


B 


Suppofe  that  the  lines  C  D  and  £  F  be  parallels  to  one  and  the  felfe  fame  line  A B 
and  let  them  not  be  parallels  the  one  to  the  other.  Then  I 
fay,that  the  two  lines  CD  &  £  £,are  diredly  fet  the  one  to 
the  other.  For  for  as  much  as  they  are  not  parallel  lines,  A 
letthemconcurreinthe  point  G,  And  from  the  point  G 
draw  a  line  cutting  the  line  AB  in  the  point  H.Now  by  the 
former  propofition  the  angles  AHG  &.HGC  are  equall 
to  two  right  angles,but  by  the  fame  propofitio,  the  angle  — 

A  H  g,is  equall  to  the  alternate  angle  H  G  f*  Wherefore  G 
the  angles  HG  C and  HG Fare  equal  to  two  right  angles. 

Wherefore  (by  the  iqpropofitionJthelinesC’G’  and  FG 
are  drawen  diredly  and  make  one  right  line.  Wherfore  al¬ 
fo  the  lines  C  D  and  £  F  are  fet  diredly  the  one  to  the  other:  and  being  produced  they 
will  make  one  right  line. 


G  E 


r 


$&The  lo.Trobleme,  The luBropoJition* 

By  a  point  geuen  Jo  draw  vnto  a  right  line  geuen  ^  a  parallel 
line . 


^^^Tppofe  that  the  point  geuen  be 
ytfind let  the  right  line geuen  be  e 
fg  q  it  is  required  by  the  point 
geuen  yiamely  Jffio  draft  Vnto  the  right 
line  ©  C?a  parallel  line.  Take  in  the  line 
C  a  point  at  alladuentures 9  and  let  the 
fame  be  D.and  (by  the firft peticio)dralb 
a  right  line  from  the  point  A,  to  the  point 
D.And  (by  the  2  3  .propofition)  Vpon  the 
right  line geuen  A  T)}and  to  the  point  in 
it  geuen  A^make  an  angle  VAE^quall  to  ^  v  0 

the  angle  geuen  A  VC.  And  (by  the  14 , 

propofition) put  vnto  the  line  yf  E  the  line  A  FdtreHly^  in  fuch  forte  that  they 

LJUi  both " 


Parallellines 
are  ^nderfian* 
ded  to  be  fro  dan¬ 
ced  infinitely. 


Gwflru&it*. 

Deimnftrathn. 


TheJirB^Boo^e 

both  make  one  right  line.  And  fora/much  as  the  right  line  A  D  falling  Vpon  the 
right  lines  ft  C  and  E  F,doth  make  the  alternate  angles ,namely,E AD tand  A 
IDC equally  one  to  the  other, ther for  e(by  theiy,propofition)EFis a  parallel 
line  toft  C.Wherfore  by  the  point  geuen  ynamely  A,is  draWne  to  the  right  line 
geuen  ftCa  parallel  line  E  A  F:  which  "teas  required  to  he  done „ 


This  proportion  is  to  be  vnderftandcd  ofa  point  geuen  without  the  line  ge- 
uen,and  in  fuch  forte  alfo,  that  the  fame  line  geuen  being  produced, doo  not  fail 
vppon  the  pointc  geuen* 

The  n.Theoreme.  The  izfPropofition* 

ffoneofthefydes  of  any  triangle  be  produced:  the  outwarde 
angle  that  it  ma\eth,is  equal  to  the  mo  inward  and  oppofite 
angles. ^And  the  three  inwar de  angles  ofa  triangle  are  equall 
totWorwbt  anvles. 


Vppofej  A  ft  C  he  a  triangle, 
produce  one  of yfides  therm 
of  namely  yC ft  to  the  point  e  ID. 

Then  Ifay ,  that  the  outWarde 
angle  A  CD  is  equallto  the  two  inward  e 
and  oppoflte  angles  C  Aft  &  AftCiandy 
the  three  inwar  de  angles  of  the  triangle , 
that  is, the  angles  A  ft  C,ftC  A,and  C  A 
ft  are  equall  to  two  right  angles.  For(by 
the  propofit  ion  going  before  )rayfe  Vpfro 
the  point  C,a  parallel  to  the  right  line  A  b 
ft, and  let  the  fame  be  C  E.Andforafmuch 
as  Aft  is  a  parallel  to  C  E,and  Vpon  them  fade  th  the  right  line  ACx  therefore 
the  alternate  angles  ft  AC  and  ACE  are  equaU  the  one  to  the  other ,  Agayne 
forafmuch  as  A  ft  is  a  parallel  Vnto  C  E,and  Vpon  them  faUeth  the  right  line  ft 
D,tber fore  the  outward  angle  EC  D  is  (by  the  zy.propofition)  equallto  the 
inward  and  oppoflte  angle  A  ft  C. And  it  is  proued  that  the  angle  A  C  E  is  equal 
to  the  angle  ft  AC:  wherfore  the  whole  outwarde  angle  A  CD  is  equall  to  the 
two  inward  and  oppoflte  angles, that  is, to  the  angles  ft  AC  and  A  ft  CJPut  the 
angle  AC  ft  common  to  them  both,VFherfore  the  angles  AC D and  A  C  fttare 
equall  to  thefe  three  angles  AftQftC  A,and  ft  AC.  ftut  the  angles  AC  Do* 
AC  ft  are  equall  to  two  right  angles  (by  the  ij.  propofition)iwherfort  the  angles 
A  C  ft,  C  ft  A,  and  C  A  ft  are  equall  to  two  right  angles .  If  t  her  fore  one  of  the 
fides  of  any  triangle  be  produced,the  outward  angle  that  it  mahthfls  equad  to 
the  two  inward  and  oppoflte  angles.  And  the  three  inward  angles  ofa  triangle 

are 


of Euclide  $  Elements s . 


Fol.^u 


are  e  quail  to  ffro  right  angles:  Tfrhicb  was  required  to  he  demonfirated, 

Euclide  demonftrateth  either  part  ofthis  compofcd  Theoreme,by  drawyng  fro 
one  angle  of  the  triangle  a  parrallel  line  to  oneofthcfides  of  the  fame  triangle, 
withoutthe  triangle.  Either  part  therof  may  alfo  be  proued  without  drawyng  of 
a  parallel  linewithoutthe  triangle,  only  chaunging  the  order  of  the  thinges  re¬ 
quired  or  conclufious.For  Euclide firftproueth  that  the  outwarde  angle  of  atri • 
angle(oneofhisfides  beyngproduced)is  equallto  thetwo  inwardeand  oppofne 
angles;  and  by  that  he  proueth  the  fecond  part:  namely,that  the  3  .inward  angles 
o  fa  triangle  are  equall  to  two  right  anglesJBut  here  it  is  contrariwile.  Forfirft  is 
proued  that  the  three  inward  angles  ofa  triangle  are  equallto  two  right  angles, 
and  by  that  is  proued  the  other  part  oftheTheoreme,  namely,  that  one  fide  ofa 
triagle  beyng  produced,  theoutward  angle  is  equal  to  the  two  inward  and  oppo¬ 
fite  angles,And  that  after  this  maner* 


Suppofe  that  there  be  a  triangle  ABC, and  produce  the  fide  BC  to  the  point  E.  And  take 
in  the  line  B  C at  al  a  point  auentures  which  let  be  F : 
&drawalinefroin^toF.AndbythepointFdrawe  A  n 

vnto  the  line^S  a  parallel  line  (by  the  former  pro- 
pofition)which  let  be  F  D,  Now  forafmuch  as  F  D  is 
a  parallell  vnto  A  B, and  vpon  them  falleth  the  right 
line  AF,and  alfo  the  right  line2?C,therfore  the  alter¬ 
nate  angles  are  equalhand  alfo  the  outward  angle  is 
equall  to  the  inward  angle.Wherefore  the  whole  an¬ 
gle  A  F  C is  equall  to  the  angles  F  A  B  and  ABF.  And 
by  the  fame  reafon(if  by  the  point  F  we  draw  aparal- 
lel  line  to  the  line  A  C)  may  we  proue  that  the  angle 
A  F  B  is  equall  to  the  angles  F  A  C,and  A  C  F . Wherfore  the  two  angles  A FB  8c  AFC 
are  equall  to  the  three  angles  of  the  triangle  A  B  C.But  the  two  ang  lesAFB  8c  AFC 
are(by  the  1 3  .prop  ofition)equall  to  two  right  angles,  Wherfore  alfo  the  three  angles 
of  the  triangle  ABC  are  equall  to  two  right  angles. 

But  the  angles  ACF  and  <^ACE  are  alfo  (by  the  13.  propofition)  equall  to  two 
right  angl  es  .Take  away  the  angle  A  C  F  which  is  common ,  w herfore  the  an  gle  remai- 
ning,namely,the  outward  angle  ACEis equall  to  the  two  angles  remaining,  namely, 
to  the  two  inwarde  and  oppofite  angles  A B  C and  C  A B ;  which  was  required  to  be 
proued. 


Eudemus  affirmeth  that  the  latter  part  ofthis  Theoremc ,  The  three  angles  ofa 
triangle  are  equall  tot\\>oright  angles, was  fir  ft  foundout  by  Jp  thagoras^whofe  demon; 
ftration  thereof  is  thus. 


Suppofe  that  there  be  a  triangle  AB  Crand  by  the  points,  draw  (by  the  former 
propofition)  vnto  the  line  £C,a  parallel  line,  which  let  beD£.And  forafmuch  as  the 
right  lines -BCand-DfEare  parallels, and  vpon  them  falleth 
the  right  lines  A  B  and./4C,therefore(by  the  2p.  propofiti-  T> 
onj  the  alternate  angles  are  equall.  Wherefore  the  angle  D 
A B  is  equall  to  the  angle  ABCran<X  the  angle  £^Ctothe 
angled  CB.  Adde  the  angle  B  A  C  common. Wherefore  the 
angles  DAB,  BA  C,CA  E ,  that  is,  the  angles  DAB  and#  B 
v4£,namely,two  angles  equal  to  two  right  angles^are  equal 
to  the  thre  angles  of  the  triangle^  B  C.  Wherfore  the  thre  angles  of  a  triangle  are  e- 
quail  to  two  right  angles:  which  was  required  to  be  proued. 

The  conuerfe  ofthis  propofition  is  thus* 

Mq.  If 


sth  other  de- 
manffratsea* 


The  latter  part 
ofthis  Theo  • 
remejirji  found 
tut  bj  IUthaga-. 
ras. 

T he  demanftra* 
tion  thereof  af¬ 
ter  him » 


The  canuerfeof 
shit  props  fit  ton. 


Demonfiratiom 
cf  the  firfl  part 
•fthe  canuerfe . 


Demonfiration : 
cf the  fecond 
part  of  the  con- 
oterfe. 


4  Cerrollarj. 


Euerj  right  li¬ 
fted  figure  is  re¬ 
fitted  in  tri¬ 
angles, 

A  triangle  is 
the fir (l  of  all fi- 
gures. 

Into  hove  many 
triangles  a  fi¬ 
gure  may  he  rc- 
feltttd. 


7  he  firfl  Boo{e 


If  the  outward  angle  of a  triangle  be  equall  to  the  tWo  inward  angles  oppofite  again  ft  it :  one  of  the 
fides  of  the  triangle  is  produced, and  the  line  without  the  triangle,  is  dravten  direttlj  to  the fide  of  the 
triangle, &  maketh  one  right  line  With  it.  And  if  the  thre  inWard  angles  of  a  rettiline figure  be  equal 
to  two  right  angles,thefkme  rettilinc figure  is  a  triangle » 

Suppofe  that  there  be  a  triangle  ABC: and  let  the  outward  angle  A  CD  be  equal  to 
the  two  inward  &  oppofite  angles  A  B  Cand  CAB. Then 
I  fay  that  the  fide  BCis  produced  to  the  poynt  D,  And 
that  -5C2)  is  one  right line,For  forafmuch  as  the  angle 
ACD  is  equal  to  the  two  inward  &  oppofite  angles, addc 
the  angle  ACB  common.  Wherefore  the  angles  ACD 
and  ACB  are  equal  to  the  three  angles  of  the  triangle  A 
B  C  .But  the  three  angles  of  the  triangle  ABC  are  equall 
to  two  right  angles  Wherefore  alfo  the  two  angles  ACD 
and  ACB  are  equall  to  two  right  angles .  But  if  vnto  a  _ 
right  line,and  to  a  point  in  the  fame  line  be  drawen  two  B 
right  lines, not  both  on  one  and  the  fame  fide,  making 
the  fide  angles  equal  to  two  right  angles  :thofe  two  rightlines  fhal  be  drawe  dire&ly, 
and  make  one  right  linefby  the  H-propofirion.^Wherefore  the  right  line  BC  is  dra¬ 
wen  dire&ly  to  cue  line  C  D,  and  fo  is  B  CD  one  right  line;  which  was  required  to  be 
proued. 

Agayne  fuppofe  that  there  be  a  certayne  reftilincfigure  AB  C,hauing  onely  three 
ang'es,  namely,  at  the  pointejM^C:  which  angles  let  be  equal  to  two  right  angles 
Then  I  fay  that  *ABC  is  a  triangle-Firll^C  wrohe right 
line .  For  draw  the  line2?  D.  And  forafmuch  4s  in  either  A 

of  the  triau  gles  A  B  D  and  D  B  C,  the  three  angles  are  e- 
qua!  to  two  right  angle  s,of  which  the  angles  at  the  points 
e//,B,C,are  equal  to  two  right  angles.  Wherefore  the  an¬ 
gles  remayning, namely,  ADB  and  CDS  are  equall  to 
two  right  angles.  Whereforef  by  the  1 4-propofition )  the 
line  D  Cis  let  dire&ly  to  the  lineD  A,  Wherefore  the  fide 
AC  is  one  right  line.And  in  like  fort  may  we  proue  that 
the  fide  zABis  one  right  line,and  alfo  that  the  fide  BCis 
one  right  line.  Wherefore  the  figure  eA  B  Cis  a  triangle: 
which  was  required  to  be  proued. 


By  the  fec5d  part  of  this  29,propofitio,  namely  fhree  angles  of  a  triangle  are  equall 
to  tworight angles,  may  ealely  be  knowen ,  to  how  many  right  angles,  the  angles 
within  any  figure  hauing  right  lines  and  many  angles  are  equall.  As  arc  figures 
offower  angles, of fiue angles,offixe  angles, and  fo  confequently:  and  infinitly* 
And  this  is  to  be  noted , that  euery  rightlined  figure  is  icfolucd  into  triangle. 
For  that  a  triangle  is  the  firfl:  ofall  figures .  For  two  lines  accomplifh  no  figures 
V  Vherfore  how  many  fides  the  figure  hath, into  fo  many  triangles  may  it  be  rc- 
folued,fauing  two.As  if  the  figure  hauefowerfides,itis  refolucd  into  two  trian¬ 
gles, if  ithauefiuefides,into5.triangles:if6  fides  into  4, triangles,  andfo con¬ 
fequently  ,and  infinitly  .And  it  is  proued  that  the  three  angles  ofeuery  triangle 
are  equall  to  two  right  angles.  VYhereforeifyou  multiply  the  number  of  the 
triangles, into  which  the  figure  is  refolued,  by  two>youfhall  haue  the  num¬ 
ber  of  tightangles ,  to  which  theangles  of  thefigure  are  equall.  So  the  angles  of 
euery  qua  irangled  figure  are  equall  to  4,right  angles.  For  it  is  compofedoftwo 
triangles.  And  the  angles  ofafiue  angled  figure  are  equal  to  6-rightangles,  for  it 
is  compofed  of  three  triangles  ,andfo  forth  in  like  order. 

The  redieft  andapteftmanerto  reduce  any  rc&ilinc  figure  into  triangles,  is 

thus 


ofEuclides  Elementes , 


Eol\i, 


thus.From  any  one  angle  of  the  figure  to  euery  other  angle  (of  the  fame)  bey  iig 
oppofite  vnto  it,  drawe  a  right  liue,fo  (hall  you  haue  all  the  triangles  of  that  fi  * 
gure  described* 

In  a  quadragle, 
from  one  angle 
you  can  drawe 
butonelyne  to 
thcoppofice  an 
gle,by  which  it 
is  deuided  into 
two  triangles 
only.  In  a  pen¬ 
tagon  figu  re, 
from  one  angle 
you  may  draw 
lines  to  two  op 
polite  angles, 
and  fo  you  fhai 

haue  three  triangles, In  an  Hexhgon, you  may  from  One  angle  draw  lines  to  thre 
oppofiteangles,and  fo  fliall  you  haue  4.  triangles. In  an  heptagon, from  one  an¬ 
gle  may  be  drawne  lines  to  foure  oppofite  angles, and  fo  fhal  there  be  flue  trian^ 
gle.  And  fo  confequently  ofthc  reft*  As  you  fee  in  the  figures  here  fet. 


This  thing  may  alfo  be  thus  cxprefled.In  any  figure  of  many  fides, the  num-  Zotnbwrhilii- 
ber  of  the  angles  of  the  figure  doubled,  is  thenuber  of  the  right  angles  to  which  ZerofrightZZa* 
the  angles  ofthe  figure  are  cquall/auing*  foure.  As  for  example*  gfa  9itfo  which 

Let  there  be  an  hexagon  figure  ABODE  F,and  within  it  take  a  point  at  all  the«”Sl“  °f  '*• 
auentures,namely,G*  And  draw  from  the  fame  point  figure*™  w 

to  euery  one  ofthe  angles  a  right  line,8c  fo  dial  there 
be  comprehended  in  the  figure  fo  many  triangles,  as 
there  are  angles  in  the  fame.  Whereforeby  this  32,. 
propoficion  all  the  angles  of  thefe  triangles  taken  toa  c 
gether,arc  equall  to  double  fo  many  right  angles,  as 
there  beangles in  rhefigure*  Wdierefoteforafmuch 
as  there  are  fixe  triangles,  there  are  tweluc  right  an* 
glcs  .Butall  the  angles  at  the  point  G  arc  equall  to  4, 
right  angles  by  the  i^propofmon.  Wherefore  rake 
away  foure  out  oftwelue,and  there  reft  eight,  Wher 
fore  thefixe  angles  in  the  Hexagon  figure  are  equall  to  eight  right  angles. 

By  that  which  hath  now  bene  declared, it  foloweth  that  all  the  angles  of  any  fi- 
gure  hauing  many  fides, take  together,  are  equal  to  twife  fo  many  right  angles,  mcZcJJZj, 
as  the  figure  is  in  the  reaw  or  order  of  figures.A  triagle  is  the  firft  figure  in  order* 

&,his  angles  are  equal  to  two  right  angles  .which  are  twife  one.  A  quadrangle  is  AtrUnsle  *he 
the  fecond  figure  in  order*  Wherfore  his  angles  are  equal  to  fower  right  angles  ** 

which  are  twife  two.The  order  offigures  is  gathered  ofthe  Tides,  For  if  you  rake  ^  ^draHle 
two  from  the  number  ofthe  fides  of  a  figure, the  number  of  thefidcs  remayning,  the  fcco”d> an* 
is  thenumber  oftheorder  ofthefigure.  As  if  you  will  know,how  many  in  order 
is  a  figure  of  fixe  fides:  from  fix(which  is  thenumber  of  his  fidcs)take  away  two*  offigures  hr* 
and  there  will  remains  foure.  VVherforca  figure  offixe  fides  is  the  fourth  figure  thsredv. 

M.u*  in 


A  B 


other  Cor - 

VelUtj. 


i  n  the  order  of  figures  .Then  double  fourc/o  {hall  you  hauc  cigh  r.  Wherefore 
the  angles  therof  arc  equal!  to  eight  right  angles. And  fo  of  alLoth  cr  figures. 


Hereby  alfoitis  manifeft, that  the  outward  angles  of  any  figure  of  many  fides 
taken  together, are  equall  to  fourc  right  anglcs.For  the  inwardc  angles  together 
with  the  outward  angles ,are  equall  to  twife  fo  many  right  angles, as  there  be  an¬ 
gles  in  the  figurc(by  the  i3.propofition)Bnt  the  inward  angles  are  equal  to  twife 
fomany  rightangles,  asthere  be  angles  in  the  figure,  fauyng  foure:  as  it  was  be¬ 
fore  declared.  VVherfore  the  outwardangles  are  always  equal  to  foure  right:  ana 
gles*  As  for  example. 

Suppofethat  there  be  a  pentagon3A  BCD  E.And  produce  the  fiue  fidcs  ther- 
-ofto  the  points  FjG^jKjL.Nowfby  thei^. 
propofitio)the  two  angles  at  the  point  A  fiiall 
be  equal  to  two  rightangles  .And(by  the  fame) 
the  two  angles  at  the  pointe  B  fhall  be  alfo  e- 
quall  to  two  right  angles*  And  fo  taking  eucry 
two  angles, they  fliaii  be  in  all  equall  to  tenne 
right  angles .  .V  Vherforc  taky ng  away  the  in¬ 
ward  angles  ,  whiche(as  hath  before  bene  pro- 
ued)  are  equall  to  fixe  righteangles,  the  out¬ 
wardangles  (hall be  equallto  fower  right  an* 
gles.Andfo  of  all  other  figures. 


other  Ctr-  ^ 15  a^°  manifeft»that  euery  pcntag5,which  is  fo  deferibed .thatech  fide  therof 

*  deuideth  two  of  the  ocherfidesjhathhisfiueangies  equall  to  two  rightangles. 

For  fuppofe  that  ABC DE :  be  fuch  a  pentagon  as  is  there  required  fo  that  let  the  fide 
AC  cut  the  fide  B  E  in  the  point  G-.&c  let  the  fide  AT)^  cut 
the  fame  fide  B  £  in  the  point  F.  Now  the  by  this  propo- 
fitiontheangle^F(7fhalbeequalltothe  two  angles  at 
thepointFand^D:namdy,theoutwardangletothetwo  ® 
inward  and  oppofite  angles. And  by  the  famereafon  the 
angle  F  G  A  is  equal  to  the  angles  at  the  points  C  and  E 
which  are  in  the  triangle  C£<?.But  the  two  angles  AFG 
and  F  G  ^together  with  the  angle  at  the  point  A,  are  e- 
quall  to  two  right  anglesf  by  this  propofition ).  Where¬ 
fore  the  fower  angles  at  thepointes,  B,C,  £>,£,  together 
with  the  angle  at  the  point  A, are  equal  to  two  right  an¬ 
gles  .-which  was  required  to  beproued. 


jtn  other  Cor- 
tralLrj. 


<An  ether  Cor- 
O'xMerjs 


By  this  propofition  alfo  it  is  manifeftjthat  cuery  angle  ofan  equilate  trian'- 
gle  is  two  third  partes  ofa  right  angle.And  that  in  a  triangle  of  two  equall  fides 
hauing  aright  angle  at  the  toppe, either  ofthe  two  angles  at  the  bafe  is  the  halfe 
of  a  right  angle.  And  in  a  triangle  called  Scalenum.fuch  a  Scalenu  ( I  fay  )  which 
is  made  by  the  drawght  ofa  perpendicular  line  from  any  one  oi  the  angles  of  an 
equilaier  triagleto  the  oppofite  fide  therof, one  angle  is  a  right  angle,an  other  is 
two  third  parts  ofa  right  angle, namely , that  angle  which  was  alfo  an  angle  of  the 
equilater  tnanglejwherforeofneceffity  the  angle  remaining  is  one  third  part  of 
a  rightangle.For  the  three  angles  ofatriaglc  mull  be  equall  to  two  rightangles* 

Moreouer  by  this  propofition  it  is  manifeftjthat  if  there  be  two  triangles, 
and  if  two  angles  of  the  one  be  equal  to  two  angles  of  the  othenthc  angle  remai- 

niog 


mng  {lull  alfo be  equall  to  the  angle  remay  mng.F or  fowCrauch  as  three  angles 
of  any  criaugle  are  equal  to  three  angles  ofany  other  triangle  (for  that  in  ech  the 
three  angles  are  equal  to  two  right  angles)- If  from  ech  triangle  be  taken  away 
the  two  equall  angles  }the  angle  remay  mng  fhall  (by  the  common  lentence)be 
equallto  the  angle  remay  mng. 


And  here  I  thinke  it  good  to  (hew  how  to  detiide  a  right  angle  into  three  c- 
quail  parteSjfor  that  the  demonftration  thereof  depended!  of  this  proportion. 


A 


Suppofe  that  there  be  a  right  angle  C^contayned  of  the  right  lilies  AB  and  B  C\& 

in  the  line  B  C,take  a  point  at  all  aduentures, which  let  be 
D.An d  vpon  the  line  B  ‘D  deferibef  by  the  firft)an  equila- 
ter  triangle  B  E>  £,And(by  the  p.propofition)deuide  the 
angle  D  B  E  into  two  equall  partes  by  the  right  line  B  F. 

Then  /  fay  that  the  right  angle  A  B  Cis  deuided into  thre 
eq  ual  parts  by  the  right  lines  B  E  and  B  F.For  forafmuch 
as  £  F  D  is  an  equilater  triangle,therfore  as  hath  before 
bene  declared,  the  angle  SB  D  is  two  thirdc  partes  of  a 
right  angle.  B  ut  the  whole  angle  ABC3  is  a  right  angle. 

Wherfote  the  angle  remaining, namely,^££  is  one  third 
part  of  a  right  angle.  Again  forafmuch  as  the  angle  EBD, 


is  two  third  partes  of  a  right  angle,andit  is  deuided  into  two  equall  parts  by  the  ri^ 
lne  B  £3therefore either  of  thefe  two  angles  E  B  E  8cF  BT)is;  one  third  part  of  a  right 
angle.Wherefore  the  three  angles -<42?  £,£2?  £  and  2)  are  equall  the  one  to  the  o- 
ther.  Wherefore  the  right  angle  A  B  Cis  deuided  into  three  equall  partes  by  the  right 
lines  B  E  and  B  F:  which  was  required  to  be  done. 


1'he  %\fTheoreme.  The  TftfPropofition* 

Two  right  lines  ioyning  together  on  one  and  the  fame  fide, 
mo  equall  parallel  lines:  are  alfo  themfelues  equall  the  one 
to  the  other >and  alfo  parallels • 

fppofe  that  A  3  and  CD  he 
right  lines  equal \  and  parallels : 
and  let  thefe  Vtoo  right  lines 
Cand  3D  ioyne  the  together > 
the  one  on  the  one  fide 3and the  other  on $ 
other  fide -Then  I  fay  that  the  lines  AC 
i?3  Dare  both  equally  alfo  parallels. 

S)raH?  (by  the  firfl  petition)  a  right  line 
from  the  point  3  to  thepoint  Ct  And fora 
afmuchasA3  is  a  parallel  toCD,  and 
\>po  them  falleth  the  right  line  3  C,  tber* 
fore  the  alternate  angles  A  3C  and  3  C 
P  are  equall  the  one  to  the  other  (by  the  %9.propofition).  And  forafmuch  as 
the  line  A3  is  equallto  the  line  C  D3and  the  line  3  €  is  common  to  them  both , 

MJiL  there* 


ffov>  t»  fUttsde# 
right  angle  int $ 
three  equall 


ThefirtlTfiookp 

therefore  thefe  Wbo  lines  AB  and  B  Cyare  equall  to  theft  tSbo  lines ) BCandC 
3j,and  the  angle  A  B  Cis  e quail  to  the  angle  B  C DyVberfore  (by  the  4.pro* 
propofiuimfihe  bafe  B  D  is  equall  to  the  bafe  A  C,and  the  triangle  A  B  Cfi  e- 
quail  to  the  triangle  B  C  T>yand  the  angles  remayning  are  equall  to  the  angles 
remay  ningthe  one  to  the  other  J>nder  Tbhich  are fub  tended  equall  fide  suffers* 
fore  the  angle  AC  Bis  equall  to  the  angle  CBD ,  and  the  angle  B  AC  to  the 
angle  B  T)  CAnd  forafmuch  as  y>pon  thefe  right  lines  A  C and  B  Dfalletb  the 
right  line  B  C  making  the  alternate  angles ,that  is  the  angles  ACB  and  CBD , 
equall  the  one  to  the  other  ^therefore  (by  the  zy.propofition)  the  line  ACisa 
par  ailed  to  the  line  B  tD.Jfnd  it  is  prouedthat  it  is  equall  Vnto  it .  Wherefore 
two  right  lines  soyning  together  on  one  and  the fame  fide  two  equall  lines  which 
are  parallels  ,are  alfo  themjelues  equall  the  one  to  the  other, and  aljo  parallels: 
Tbhicb  'toas  required  to  be proued. 

The  i^.Theoreme.  T he  34.  Tropo/ttion. 

fn  farattelogrammesjhefides  and  angles  which  are  oppofae 
the  one  to  the  other ,  are  equall  the  one  to  the  other*  and  their 
diameter  deuideth  them  into  two  equall  partes. 

BVppofe  that  A  BOD  be  a  parallelogramme  and  let  the  diameter  ther * 
of  be  B  C.Tben  I  fay  that  the  oppofitefides  and  angles  of  the  paralhlo » 
gramme  ACDB  are  equall  the  one  to  the  other yandj  the  diameter  BC 
deuideth  it  into  Wbo  equall  partes .  For  forafi 
much  as  A  B  is  a  parallel  line  bnto  C  fyand  V* 
pon  them  falleth  a  right  line  B  C:  therfore  (by 
the  2  <y,propofition)the  alternate  angles  ABC 
and  BCD  are  equall  the  one  to  the  other .  A • 
gay  ne forafmuch  as  AC  is  a  parallelline  to  B 
lD,and  Vppon  them  falleth  the  right  lyne  B  C: 
therfore  (by  the  fame)  the  alternate  angles, 
that  is, the  angles  ACB  and  CBD  are  equall 
the  one  to  the  other .  Nolb  therfore  there  are 
tV>o  triangles  ABC  and  BCDfhauing  Wbo  an - 
gles  of the  oney  namely %  the  angles  ABC  and  ACB  equall  to  Woo  angles  of the 
other  that  isyto  the  angles  BCD  andC  B  D,  the  one  to  the  other jmd  one fide 
of  the  one  equal  to  one  fide  of  the  otherjiamelyjhatfydethatlietb  beWbene  the 
equall  angles  Jbhich  fyde  is  common  to  them  both, namely, the fide  B  C,  Wber* 
fore  (by  the  26.  proportion)  the  other  fides  remaining  are  equall  to  the  other 
fidesremamingghe  oneto  the  other ,and  the  angle  remaining^  equal  to  the  an* 
gle  rmaynmgyVhexfore  the fide  A  B  is  equall  to  the  fide  CD,md  the  fide  AC 


ofEuclides  Ekmentes . 


to  the fide  B  Dgs*  the  angle  B  AC  is  equal  to  the  angle  B  D  C.And  forafmucb 
as  the  angle  ABCis  equal  to  the  angle  B  C  Dtand  the  angle  CBDto  the  an  * 
gle  AC  B :  tberfore  (by  the  fecond  common  f entente)  the  whole  angle  A  B  D 
is  equall  to  the  whole  angle  A  C  Dt  And  it  is  proued  that  the  angle  B  AC  is  e* 
quail  to  the  angle  C  D  BjAClurfore  in  parallelogrammes }  the fides  and  angles 
which  are  oppojite  are  equall  the  one  to  the  other,  I  fay  al(o  that  the  diameter 
therof deuideth  it  into  two  equall  partes. For  forafmuchas  A  B  is  equall  toCD} 
and  BCis  common  to  them  bothgherfore  thefe  two  A  Band  BCare  equall  to 
thefe  two  BCandCDtandthe  angle  A  B  Cts  equal  to  the  angle  BCDyFher 
fore  (by  the  ^.propofition)  the  bafeACis  equall  to  the  ba  fe  B  D>and  the  t*i* 
angle  A  B  Cis  equall  to  the  triangle  B  C  ID.  VFherfore  the  diameter  B  C  den 
uide  tb  the  psralldogramme  A  B  CD  into  two  equall  partes :  which  is  all  that 
Was  required  to  be  proued. 


In  this  Theorerae,are  demon  Crated  three  paffions  or  properties  ofparallei? 
logrammes.  Namely, that  their  oppofite  files  are  equall;  that  their  oppofice  an? 
gles  are  equall-.and  that  the  diameter  deuideth  the  parallelogrammc  into  two  e- 
quall  partes*  Which  is  true  in  all  kindes  ofparallelogrammes.  There  arc  fo  wer 
kindes  of  parallelogrammes  ,a  iquare,  a  figure  of  one  fide  longer  then  the  other, 
a  Rhombus,or diamond  figure,and  a  Rhomboides  or diamondhke figure.  And 
here  is  to  be  noted, that  in  thofe  parallelogrammes,  all  whofe  angles  ar  right  an- 
gles(as  is  a  fquare,and  a  figure  on  theone  fide  longer)  the  diameters  do  notonly 
deuide  the  figure  into  two  equall  partes,but  alfo  they  are  equal  the  one  to  the  o* 
ther*As  for  example. 

Suppofethat^SCD  beafquare,  4  B  A  B 

or  a  figure  on  the  one  fide  longer, and 
draw  in  it  thefe  diametres  A  D  and  5 
C.And  forafmuch  as  the  line  >4  5  is  e- 
quallto  the  line  CD  (by  the  definitio 
of  a  fquare,and  of  a  figure  on  the  alfo 
one  fide  loger)&  the  line  A  Cis  com¬ 
mon  to  the  both  ;  therfore  two  fides 
of  the  triangle  A  5  Care  equal  to  two 
fides  of  the  triangle  A  CD  gat  one  to 
the  other,  and  the  angles  which  they  a 
contayne  are  equall,  namely,  the  an-  C  v 

gles  5  AC  &  AC  Z>,  for  they  ate  right  angles.  Wherefore  the bafes  namely,  the  diame¬ 
ters  A  D  and  5C,arefby  theq.propofitionjequal, 

Butin  thofe  parallelogrames  whofe  angles  arcnotrightangles,as  is  a  Rhom¬ 
bus  ,and  a  Rhomboides, the  diameters  be  euer  vnequall.  As  for  example. 

Suppofe  that  ABC  D  be 
a  Rhombus,or  a  Rhombaides 
and  drawe  in  it  thefe  diame¬ 
ters  A  C  and 5  D.  And  foras¬ 
much  as  >45  is  equall  to  C2>, 
and  B  Cis  common  to  them 
both,&the  angle>45  Cis  not 
equall  to  the  angle  5  C  D  (  by 
the  definition  of  a  Rhombus 
and  alfo  of  a  Rhombaides) 

M.iiii. 


B 


T  tire  fastens  of 
far  alio  tegrames 
demo  ft  rated  in 
this  T horeme. 
Power  kjndes  of 
faraHelo- 
grammet. 


’  ■  ‘-f-. 


there 


*The  eonuerfe  of 
this  prep  opt  ion 
itfter  Vrqclm, 


A  Corollary  ta¬ 
ken  emt  of  v’ 


ThefirslTiooke 

therefore  (by  the  24,propofition)  the  bafes  alfo  are  vnequall,  namely,  the  diameters 
zACa.n6.BD. 

Agaync.Inparallelogrammesofequallfides,asareafquare,  anda  Rhom* 
bus, the  diameters  do  notonely  delude  the  figures  into  two  equall  partes,  but 
alfo  they  deuide  the  angles  into  two  equall  partes* 

For  fuppofe  that  there  be  a  fquare  or  Rhombus  AB  CD,  and  draw  the  diameter  e/tf 
D. And  forafmuch  as  the  fides  A  B  and  B  D  are  e- 
quall  to  the  fides  A  C  and  CD(for  the  figures  are 
equilateral  and  the  angles  sAB  D  and^  CD  are 
equally  for  they  are  oppofite  angles )  and  the  bafc 
ADisrommontoboth  triangles.  Therefore  (by 
the  fourth  propofition)the  angles  BAD  &  CAD 
are  equail,and  fo  alfo  are  the  angles  BDA  and  C 
‘D  A  equall.  Wherfore  the  angles  i?  AC  and  CD  B 
are  deuidedinto  two  equall  partes. 


Butin  parallogratnmes  whofe  fides  arenot  equall,  fuch  as  area  figure  on  the 
one  fide  longer, and  a  Rhomboides  it  is  not  fo. 

For  fuppofe  AB  CD  to  be  a  figure  on  the  one 
fide  longer  or  a  Romboides,  And  draw  thedia-  A  B  IB 

meter  zA  D,And  now  if  the  angles  B  A  Cand  CD 
*S,be  deuidedinto  two  equall  partes  by  the  dia- 
meter-^D,theu  forafmuch  as  the  angle  CAD 
is  (by  the  2<?.propofition,)  equall  to  the  angled 
*D  .8,  the  angle  alfo  BAD  fhal  be  equal  to  the  an¬ 
gle  A  D  B(by  the  firft  common  fentence),  Wher¬ 
fore  alfo  the  fide is  equall  to  the  fide  B  Dfby 
the  tf.propofitio  ).  But  the  fayd  fides  are  vnequal: 
which  is  impoifible.Wherefore  the  angles  B  AC  c 

and  C  D  B  are  not  deuided  in  to  two  equall  partes* 

The  eonuerfe  of  the  firft  and  fecond  part  of  this  propofition  after  Proclus. 
if  a  reel ihne figure  whatfoeuer  haue  his  oppofite  fides  andangles  equall :  then  is  aparallelograme. 


For  fuppofe  that  AB  CD  be  fuch  a  figure,namely,which  hath  his  oppofite  fides 
and  angles  equall.  And  let  the  diameter  thereof  be  zAD. 

Nowforafmuchas  the  fides  A  rB  and  2D  are  equall  to 
to  the  fides'D  C  and  zA  C,  and  the  angles  which  they  co~  A_ 
tayne  are  equall,  and  the  bafe  AD  is  common  to  ech  tri- 
angle,thereforef  by  the  4,propofition  Jthe  angles  rema y- 
ning  are  equall  to  the  angles  femayriing,vnder  which  are 
fubtended  equal  fides.Wherfore  the  angled  AD  is  equal 
to  the  angle  A  D  C, and  the  angle  zA  D  B  to  the  angle  CA 
D. Wherefore  (by  the  27.  propofition )  the  line  ABisa 
parallel  to  the  line  CD^and  the  line  AC  to  the  line  B  D, 

Wherefore  the  figure  AB  CD  is  a parallogramme:which 
was  required  to  be  proued. 

\  /  \  !  ^ 

A  Corrollary  taken  out  of  Fluffates. 
tA  right  line  cutting  a  parallelogramme  Which  Way  foeuer  into  two  equall  partes ,  Jhall  alfo  de¬ 
uide  the  diameter  thereof  into  two  equall  partes. 

For 


of Euclides  Elementes.  FoL  4.?. 


For  if  it  be  polfible  let  the  right  line  G  C  deuide  the  parallelogramme  A  SR  JO  into 
two  equal  partes,  but  let  it  deuide  the  diameter  D  E  into  two  vnequal!  partes  in  the 
point  h  And  let  the  part  /  £  be  greater  then  the  part  /  Z>*And  vnto  the  line  ID  put  the 
line/O  equall(by  thg.  propofitioj.  And  by  the 
point  0,draw  vnto  the  lines -4  2>  and-#  S  apa- 
rallelline  O  £(by  the  3  1  .propofition .)  Where¬ 
fore  in  the  triangles  EG  I  and  CD  I,  two  angles ; 
of  the  one  are  equal  to  two  angles  of  the  other, 
namely, the  angles  IOF  and  I  DC  (  by  the  2p* 
propofition),&  the  angles  FI  O  8cCID( by  the 
1 5  .propofitio),&  the  fide  ID  is  equal  to  the  fide 
I O. Wherefore  (by  the  2  ^.propofition ^the  tri¬ 
angles  are  equall.Wherefore  the  whole  triangle 
Sigis  greater  then  the  triangle  D IC.  And  forafmuch  as  the  trapefium  GBDCis  fup- 
pofed  to  be  the  haife  of  the  parralleIograme,and  the  halfe  of  the  fame  parallelograme 
is  the  triangle  EB  D  (by  this  propofition)  .From  the  trapefium  GB  DC  and  the  trian- 

fle££-Dwnichare  equalhtake  away  the  trapefium  (?£  D/whichis  common  to  them 
oth,and  therefidue  namely,  the  triangle  D  IC lhalbeequall  to  the  refidue, namely, 
to  the  triagle  £/(7:butitis  alfo  leflefas  hath  before  ben  proued): which  is  impoffible. 
Wherefore  a  right  line  deuiding  a  parallelogramme  into  two  equal!  partes,  {ball  not 
deuide  the  diameter  thereof  vnequally.  Wherefore  it  fhall  deuide  it  equally  ?  which  was 
required  to  be  proued. 

An  addition  of  P  elitarius* 

Betty enettyo  right  lines  being  infinite  and  making  an  angle  geuen  :  to  place  a  line  equall  to  a 
line  geuen, in jack forte, that  it Jhau  make  Veith  one  oft  ho/e  lines  an  angle  squall  to  an  other  angle ge* 
utn,N°ty  *t  behoueth  that  the  ttyo  angles geuen  be  le/fc  then  ttyo  right  angles. 

Suppofe  that  there  be  two  lines  A  B and  A  C,making  an  angle  geuen  BA  C:  and  let 
them  be  infinite  on  that  fide  where  they  open  one  from  the  other*  And  let  the  line 
geuen  beD,  and  let  the  other  angle  geuen  be  £. 

And  let  the  two  angles  A  and  £  be  lefle  then 
two  right  angles  (  otherwife  there  coulde 
not  be  made  a  triangle,asitis  manifeft  by  the  17 
propofition) .  It  is  required  betwene  the  lines  csf 
B  and  AC  to  place  a  line  equall  to  the  line  geuen 
D, which  with  one  of  them  as  for  exaple  with  the 
line  A  C, may  make  an  angle  equal  to  the  angle  ge  ^ 
uen  £.Now  then  vpon  the  line  C  and  to  the 
pointinitcxf.make  an  angle  equail  to  the  angle 
geue£(by  the  2  3  .propofition),  which  let  be  CA 
£,And  produce  the  line  FA  on  the  other  fide  of 
the  point  A  to  the  point  G:znd  let  A  (/be  equall 
to  the  line  geuen  D  ( by  the  3 .  propofition ).  And 

by  the  point  (?,draw(by  the  3 1  .propofition)  a  parallel  line  to  the  line  oAC,  which  let 
be  G  H, and  produce  it  vntil  it  concurre  with  the  line  A  B:  which  concurfe  let  be  in  the 
point//. And  agayne  by  the  point// draw  the  line  HK parallel  vnto  the  line  (//.-which 
let  cut  the  line  exf  C  in  the  point  ZT.Then  I  fay  that  the  line  H  K  is  placed  betwene  the 
lines  AB  8c  AC  &  is  equall  to  the  line  D*And  that  the  angle  at  the  point  K  is  equall  to 
the  angle  geuen  £.Forforafmuchas(byconftru&ion)-4<?///:is  a  parallelogramme 
the  line  K  His  equall  to  the  line  A  (/(by  this  propofition ).  Wherefore  alfo  it  is  equall 
to  the  line  D.  And  forafmuch  as  the  line  A  ZTfalleth  vppon  the  two  parallel  lines,  F  G 
and  K //, therforc  the  angle  A  K His  equal  to  the  angle  F  A  K( by  the  2p.propofitio.) 
for  that  they  are  alternate  angles-Wherfore  alfo  the  fame  angle  at  the  point  £is  equal 
to  the  angle  geuen  S.  Wherefore  the  line  Z/K  being  placed  betwene  the  two  lines  AB 
and  <sA  C,and  being  equall  to  the  line  D,maketh  the  angle  at  the  point  K  equall  to  the 
angle  geuen  £:  which  was  required  to  be  done. 

Though  this  addition  of  Pclitarius  be  not  fo  muche  pertayning  to  the 

N.i*  propo- 


Hemon/lratim 
leading  to  tt  ft 
ah [nr dy tie. 


An  addition  of 
Vclitarittt. 


CenJlruHsm. 


Demmftrarim 


Demtnftrdthn 


Three  cafes  in 
this  fropofttion„ 
Thcfirjt  cafe. 


propofitiomy  ctbecaufeit  is  witty  and  femcth fomewfm  difficult,  I  thought  it 
good  here  to  ancxe  it. 

4  if  '  —*'•  l  ■  ■  .  *  .  f  44J,.  v r 

.  Theiy  .Theorems.  The  tf.Tropofoion. 

TaraUelogrammes  confi/lingvppon  one  and the fame  bafe, 
and  in  the felfe fame  parallel  lines,  are  e  quail  the  one  to  the 
other.  - 


p  Vppofe  that  the/e  paraUelo * 
f if  grammes  A  'B  CD  and  EB  4 
CF  Jo  confeft  vpon  one  and 
the  fame  bafe /hat  iss  Vppon 


& 


J  J  3  1  If 

B  Ctand  in  the  felfe  fame  pa  rallel  lines  y 
that  is  J  F,and  B  C.Tbeti  1 fay /hat  the 
parallelograrne  ABC  Bis  equal  to  the 
parallelograrne  EEC  F.For forafnuch 
as  A  $  CB  is  a  pardUetagramme,  ther* 
fore  (by  the  $  4 .propofttion )  the  fde  A 
&7is  equallto  the  fideB  C,  and  by  the 
fame  reafm  alfo  the  fide  EE  is  equall  to 
the  fide  BC/vhcrfore  JED  is  equall  to 
EFandB  Eis  common  to  them  both- 
Wherfore  the  tbbole  line  A  E  is  equall  to  the  whole  line/D  FsAnd  the  fide  Al 8 
is  equall  toy  fide  D  Cypher  fore  the/e  two  EA  and  AB  are  equall  to  theft  Vtoo 
HD  and  DC /he  one  to  the  otheriandy  angle  FDC  is  equall  to  the  angle  EABy 
namely/he  outward  angle  toy  inlrard  angle (by  y  n)tpropofitio):ts>herfore(by  jt 
4  propofition)the  bafe  E  B  is  equall  to  the  bafe  ECy  and  the  triangle  EA  Bis 
equallto  the  triangle  FED  C.Take  ale  ay  the  triangle  EDGE,  lehich  is  common 
to  them  botb.VVherefore  the re  ft  due, namely, the  trapefium  A  BG  Bis  equall 
to  the  re/idue,  that  is  Jo  the  trapefium  EG  C  FfPut  the  triangle  GB  C  commo 
to  them  both  Wherefore  the  to  hole  paralleUgramme  ABC  Bis  equall  to  the 
"tobole  parallelogramme  EB  C  FVVherefore  paralletogrammes  con f fling  vp* 
on  one  and  the  fame  bafe, and  in  the  felfe  fame  parallel  lines  yare  equal  the  one  to 
the  otber.'fchich  was  required  to  be  demonflrated. 


Parallelogrammcs  are  fayde  to  be  in  the  felfe  fame  parallel  lines, when  their 
bafes,andthe  oppofite  Hies  vnto  them,  are  one  and  the  felfe  fame  lynes  wy  th 
the  parallels* 

In  this  propofition  are  three  cafes*  For  the  line  B  E  may  cutte  the  line  AF, 
either  beyond  the  point  D,or  in  the  point  D,or  on  this  fide  the  point  D  *  When 


ofEuclides  Elementeu 


FoL\6. 


itcnttcth  the  line  A  F  beyond  the  point  D  the  demontetion  before  put  fcr^ 

ueth*  V:  .•  e 

•  ■  '  •••  -  -  :  •  .  *  \  *  *  V  i  \  '  *  > 

Butiftheline££do  cutte  the  line  AF in 
the  point  D,then  forafmuch  as  (  by  the  former 
propofition)the  triangle  £  CED  or  2?  C£  is  the 
halfe  of  either  of  thcfe  paralelogrammes  ABC 
‘Da.ndEBCF  (  for  intheparallelogramme^ 

B  CT>  the  diameter  B  D  maketh  the  triangle  B 
DCthe  halfe  of  the  fame  paraIlelogramme,and 
in  the  parallelogramme  EBCF  the  diameter 
£  Cor  DC  maketh  the  felfe  fame  triangle  £‘D 
Cthe  halfe  of  the  parallelogramc££  CF  Jther* 
fore(by  the  7. common  fentence)thcparallelo- 
grammes  A  B  CD  and  £  B  CF  are  equall. 


But  if  the  Hue  BE  do  cutte  the  lyne  *AF 
on  thi5  fi.de  the  point  D,  then  forafmuch  as  ei¬ 
ther  oF  the  lines  AD  and  EFis  equall  to  the 
line  B  C,therefore  by  the  fir  ft  common  Sentence 
they  are  equall  the  one  to  the  other.  Wherefore 
taking  awav  ED,  which  is  common  to  both y 
the  refid  ueV^  £  fhalbe  equall  to  the  refidue.D 
F.  Agaync  forafmuch  as(by  the  ^.propofitio) 
the  line  zA  B  is  equall  to  the  line  C  D,  and  (by 
the  17.  propofition)theangle£  A  B  is.equaUo 
the  angle  FDC:  therfore  (by  the  4.  proposi¬ 
tion)  the  triangles  E  2?  A  and  F  C  D  are  equal* 
Adde  the  trapefium  CDEB  common  to  them 
both  :  and  fo  (by  the  feConde  common  Sen¬ 
tence  jthe  two  paraflelogrammes  */{ B  CD  and 
££C£fhaIbe  equall  :  which  was  required  to 
beproued. 


The/ecmdeitfi* 


The  third ca ft. 


h&'The i6:Theoreme.  'ThetfSPropofition. 

ParaUelogrammes  confi fling  vpon  equall bafes,  and  in  the 
felfe  fame  parallel  lines  ^are  equall  the  one  to  the  other. 

Vppofe  that  thefe  parallelogramtnes  AB  CD  and  EE  G  H  do  con  ft jl 
ypon  equall  ba/es3that  is^pon  BCandF  G,and  in  the  felfe  fame  paraU 
■  lei  lines  9t  bat  is  3A  Hand  I B  G,  Then  I  Jay  %t  bat  the  parallelogramme  A 
BCD  is  equall  to  the  parallelogramme  EFG  FLDra'Se  a  right  line  from  the  C9”J?ruaio*‘ 
point  B  to  the  point  Eyand  an  other  from  the  point  C  to  the  point  H.  Jndfor*  Demorfr#** 

afmuch  asBC  is  equall  to  F GJhut  F  G  is  equall  to  E  H,  therfore  <B  Calfo  is  e* 
quail  to  E  H3  and  they  are  parallel  lines ,  and  the  lines  B  E  andC  B  ioyne 
them  together :  hut  npo  right  lynes  ioynyng  together  tlt>o  equall  right 

Hdj*  lines 


“V 

■l 


Three  cafes  in 
this  propofttion. 

The fir  ft  cafe. 
Eaerj  cafe  maj 
happen  fetiea 
timers  wajet. 

a. 

3» 


%  •*».*.  <f7 
•Ji-aff.’.r' 


E 


H 


7 

:  C  A*1 

y 

’  /  ■  : 

- 

Jr 

/ 

sj 

/  '1 
--  *. 

if 

•-  ■  1 

if 

7 

if 

...  .  '  .  '! 

/  j 

/ 

■  ;i 

Jr 

/ 

j 

fine s  being parallels;  art  themfeluts 
alfo  (by  the  ^  proportion)  e quail 
the  one  to  the  other ,  and  parallels, 

VVherforeEB CH  is  a parallelo* 
gramme  3and  is  equall  to  the  parallel 
lograme  ABCS):  for  they  haueboth 
one  and)  fame  bafeythat  tsfB  C,And 
are  iny  felfe  fame  parallel  lines ythat 
is3BC&  EHtAnd  byy  fame  reafoit 
alfo  the parallelograme  EFG H  is  e* 
qual  to  the  parallelograme  E  B  CH ; 

VVber fore  the  parallelograme  AB 
CS)  is  equal  to  the  parallelograme  E  &  c:  ^ 

FGH,  VVher  fore  parallelogr  antes 

conjifting  vppon  equall  bafes3andin  the  felfe  fame  parralkl  lines,  art  t quail  the 
one  to  the  other.Tahich  'toas  required  to  beproued. 

i,.  .?•  -.  .  ,  j .  ^  ^  ,  _  f.  (  , .  (  „ 

In  this  propoficton  alfo  are  three  cafes.For  the  equall  bafes  may  either  be  v  t¬ 
terly  feperated  a  fonder:  or  they  maytouefoeat  oncof  the  codes:  ortheymay 
haue  one  part  common  to  them  both. 

Ettclides  demonftration  ferueth  when  the  bafesbe  vtterly  feperated  a  fonder* 
Which  yet  may  happen  feuen  diners  waycs.For  thebafes  being  feperated  af5*> 
der, their  oppofite  fides  alfo  may  be  vtterly  feperated  a  fonder  beyond  thepoinfi 
D,as  the  Tides  A  D  and  E  H  in  the  firift  figure. 

Or  they  may  touche  together  in  one  of  the  endes,  and  the  vhole  fi  demay  be 
beyond  the  point  D,as  the  Tides  A  DandE  H  do  in  the  fecond  figure. 

Or  one  part  may  be  beyond  the  point  D,  andan  other  part  common  to  them 


4 


b  cp 


d  B  C  F 


<3  3  C  F 


both  ,a  s  in  the  third  figure,the  fides  A  D  and  E  H  haue  the  part  E  D  common  to 
them  both* 

Or  they  may  iuftly  agree  the  one  with  th'cother,that  is,the  pointes  A  and  D 
may  fall  vpon  the  pointes  Eand  H:  as  in  the  fourth  figure. 

Or  the  fide  A  D  being  produced  on  this  fide  the  point  A}part  ofthe  oppofite  y. 

fide  vnto  the  bafeF  G  m'ay  be  on  this  fide  the  point  A,  and  an  other  part  may  be 
common  with  the  line  A  D,as  in  the  fifth  figure* 

Or  one  ende  ofthe  fide  EH  may  light  vpon  the  pointcA3  and  the  whole  fide  6. 
on  this  fide  of  it:  Asinthefixt  figure. 

Or  the  faid  fide  E  H  may  vtterly  be  feperated  a  fonder  on  this  fide  the  pointc 
A, as  in  the  feuenth  figure. 


And  the  two  other  cafes  affo  may  inlike  mancrfiauc  feuen  varieties:  as  in 
th„e  figures  here  vnderncth  and  on  the  other  fide  of  this  leafefet  it  is  inanifeft. 
and  here  is  to  be  noted^thac  in  thelft  three  cafes  and  in  all  their  varieties  alfo, the 
conftru&ion  SC  demonftration  put  by  Eqclidef  namely  ^thc  drawing  of  linos  fro 
the  point  B  to  the  point  E  Sc  from  the  pointe  C  to  the  point  H,and  fo  prouing 
ic  by  theforinetpropofition)  will  feme  oncly  in  the  fourth  varietie  ofech  cafe, 
there  nedeth  no  farther  conftructiomfor  that  the  conclufion  ftraight  way  folio? 
weth  by  the  former  propofition. 


The  like  *varie* 

tj  in  ech  ofthe 
other  two  cafes , 
Each  da  con - 
Jirullian  and 
demonftration 
feructhinall 
thefe  cafes,  and 
in  their9aritiet 
alfo. 


A  "DE  H 


&  F  C  <3 


A  B  D  H 


AR  T>ff 


3  F  C  <3 


»  F  O  G 


Canftntftion, 


Demonftratiott. 


i 1  '■  Cj  .  ;  l\ 

TheijaTheoreme .  7 be  ty.Tropo/ition. 

■  ",  ■’  r  ‘  i  \  ,  **  41  .  i 

T riangles  con f fling  upon  one  and the  felfe  fame  bafe?and in 
the felfe fame  par allesiare  e  quad  the  one  to  the  other* 

Vppofe  that  the  re  triangles  A  BCandlDBC  doconfift 
Vpon  one  and  the  fame  bafe>namdyi  B  Cyand  in  the  felfe 
fame  parallel  lines  jhat  is ,  jiT>  and  B  C.  Then  I  fayy 
that  the  triangle  ABCis  e quail  to  the  triangle  B(DC. 
Produce  (by  the  2.  peticion )  the  line  At)  on  ech  fide  to 
the  pointer  E  and  F.  And  (by  the  31,  proportion)  by  the 
point  Hydras  Ynto  the  line  C  A  a  parallel  line  B  E  and 
(by)  fame)  by  the  point  C,  draw  Ynto  the  line  B  !Da  pa* 
rallel  line  C F Wherefore  EBCA. 
and  D  B  C^are  paraHelogrammes „ 

And  the  paralklogramme  E  B  fAy 
is  (by  the  $  5.  propofition)  equal/  to 
the parallelogramme  (DBCF ,  For 
they  confift  Yppon  one  and  the  (elfe 
fame  bafe ,  namely ,  BCy  and  are  in 
the  felfe  fame  parallel  lines ,  that  isy 
B  (  and  E  F,  But  the  triangle  ABf 
id  (by  the  $4. propofim)tbt  halfe  of 
the  parallelogramme  EBCAyfor 
the  diameter  AB  deuideth  it  into  two 
equall  par ts:&( by  the  fame)thetri •  £  C 

angle  BBC  is  the  halfe  of  the  pa a 

rallelogra mmelDBC  F for  the  diameter  B>  Q deuideth  it  into  two  equall parts: 
but  the  balues  ofthinges  equallarealfo  equall  the  one  to  theotber  (by  they . 
common  fentence  )  therefore  ehe  triangle  ABC  is  equall  to  the  triangle  (DB 
(V  therefore  triangles  confining  Ypon  one  and  the  felfe  fame  bafe3and  in  the 
felfe  fame  parallelsiare  equal l  the  one  to  the  other :  Tphicb  Y>as  required  to  be 
demonilrated. 


Thofe 


ofEuclides  Elementes . 


FoL^.8* 


Thofc  triangles  are  faide  to  be  comayned  rithin  the  felfe fame  parallel  lines,  HowtrUngtes 
which  hauing  their, bafes  in  oneofths  parallel  lineSjhane  their  toppes  in  the  %e/hffiife°fJLe 

Other.  '  •  1  parallel  lines, 

Hereas  I  prom  i  fed  will  I  flicw  out  of  PCocIas  the  com  pari  fori  of  two  triad-  comparifonof 
gles, which  hailing  their  hies  equal!  'jti&ttc  the  Safes  and  angles  at  the  toppe  vucj  two  triangles 
quail*  Arid  Aril:  I  fay  that  the  vneqnall  angles-  at  the  toppe  being,  equall  to  two  wh°^^tie!  btl‘ejr 
right  angles, the  triangles  flialbe  cqualLAs  for  example, ,  -  ^ffJand  angles 

at  the  toppe  art 

Suppofe  that  thefe  two  triangles  ABC  and  DEF  haue  twofides  ofthe  onemarne- 
ly,  A  B  ahd  A  C^equall  to  two  fides  of  the  Other,  namely,  to  D  £  and  D  F,  eche  to  his 
correfporideni:  fide, that  is,  ABtoD  £,and  ACtoD  F, and  let  the  bafe  B  C  be  greater  v/hen  the  two 
then  the  bafe  £  F :  andfet  the  angle  at  the  point  tA  be  greater  then  the  angle  at  the  angles  at  the 
pointD.Butletthefaydeanglesatthe  pointes  ^4andD,  \  toppes  are  ec^uaU 

be  equall  to  two  right  angles. Then  I  fay  that  the  triangles 
ABC  andD  ££  are  equall. For  fora/rauch  as  the  angle's 
AC  is  greater  then  the  angle  £  D  £,vpon  the  line  £  D,and 
tothepointD  defcribe  (by  the  23.  propoikion) an  angle 
equall  to  the  angle  £^TC;which  let  be  E  D  G :  and  put  the 
line  D  G  equall  to  the  line  A  Ci  and  draw  a  line  from  E  to 
G,and  an  other  from  F  to  G;  and  produce  the  lines  ED& 

F  D  beyond  the  poynt  D  to  the  pointes  H and  AT.Now  for 
afmuch  as  the  angle  'B  AC  is  equall  to  the  angle  ED  G* 
and  the  angles  b  A  Cani  E  D  F  are  equall  to  two  right  am 
gles,thereforethe  angles  EDG  andEDF  are  equall  to 
two  right  angles  .But  the  angles  EDG  and  K  D  G,arealfo 
equal  to  two  rightangles.-take  away  che  angle  FDG  com¬ 
mon  to  them  both:  wherefore  the  angle  remayning  EDF 
is  equall  to  the  angle  femayning.G  D  K. But  the  angle  ED' 

F  is  equall  to  the  angle'H  D  K  (  by  the  1 5  .propofition)for 
they  are  hed  angles.  Wherefore  theangle  G  D  K  isequall 
to  the  angle  HDK  .And  forafmuch  as  in  the  triangle  G  D  F  the  outward  angle  G  D  H 
is(by the  3 2. propofition) equal  to  the  two  inward  and  oppofite  angles  at  the  points  G 
and  £:  which  two  angles  alfoare(by  the  5. propoiition)  equall  the  one  to  the  other: 
for  the  line DG  is  by  confku&ion  equall  to  the  line  c AC,  namely,  to  the  line  DF. 

Wherefore  the  angle  G  D.H  is  double  both  to  the  angle  at  the  point  <?,and  to  the  an¬ 
gle  at  the  point  P.Biit  the  angle  G  D  His  alfo  double  to  the  angle  GDK  (forthe  an¬ 
gle  G  D  K  is  proued  to  be  equal!  to  the  angle  K  D  H)  wherefore  the  angle  at  D  G  F 
is  equal!  to  Che  angle  G  D  K  :  and  they  zie  alternate  angles.  Wherefore  (by  the  27. 
propoiition)  thelineD  E  isaparallel  to  the  line  £  £7.  Wherefore  the  triangles  GDE 
and  FD  E  arevppon  one  and  the  felfe  fame  bafe,  namdy,  Z>£,  and  in  the  felfe  fame 
parallel  lines  D  E  and  GF  .Wherefore  by  this  propofition  they  are  equall.  But  the  tri¬ 
angle  GDE is  by  conftm&ion  equall  to  the  triangle  ABC.  Wherefore-  alfo  the  tri¬ 
angle  D  E  F  is  equall  to  the  triangle  zABC :  which  was  required  to  be  proued. 

But  now  let  the  ariglesA  A  C  and  £  CD  £  be  greater  then  two  right  angles  :  &  let  When  thej  ats 
the  angle  at  the  point  A  be  greater  then  the  angle  at  the  point  D,  as  it  was  before  The  ireater  th'**e 
I  fay  that  the  triangle  ABCis  lelfe  thea-the  triangle  £>£  £.  Let  the  fame  conlfruaion  r,ght  angles. 
beherothat  was  in  the  former.  And  forafmuch  as  the  angles  B  ACznd  S'DF  that  is, 
the  angles  EDCj  and  £CD£are  greaterthen  tworight  angles,  but  the  angles  EDG 
and  GD  A" are  equall  to  two  right  angles:  take  away  the  angle  FD  G  which  is  common 
to  them  both .  Wherefore  the  angle  remayning,namely,£D£  is  greater  the  the  amrie 
remayningpiamelyjthen  GD  A:  thatis,the 'angle  JCD  H (which  by  the  1  5.  propofition 
is  equall  to  the  angle  £  D  F)is  greater  then  the  angle  G  D  K,  wherefore  the  angle  G  D 
H  is  more  then  double  to  the  angle.  G.D  K :  but  the  angle  f?  D  His  double  to  the  an- 

N.iii;‘.  glc 


Whettthcj  *rt 
leJJ'e  then  tn>a 
right  angle t. 


cHbefirH(Boo^e 

glc  DGF ,  as  was  before  proued,  Wherefore  the 
angle  GDK is  leffe  then  the  angle  DGF.  Vnto  the 
angle  GDK  put(by  the  a  3  .propofition )  the  an¬ 
gle  OG  £  equall:  and  produce  the  line  G  L  till  it 
concurre  with  theline  ££inthepointe  L.  And 
draw  a  line  from  D  to  L.  Wherefore(by  the  2  7, 
propofition)G*  L  is  a  parallel  line  to£>  £,forthat 
the  alternate  angles  D  G  £and  G  D  K are  equal. 

Wherfore  the  triangles  G  D  E  and  LD  E are  (by 
this  propofition  Jeq  ual(for  they  confift  vpon  one 
and  the  lelf  fame  bafe,namel|y,2)£,  and  are  in  the 
felfc  fame  parallel  lines,namely,£  D  and  G  £)But 
the  triangle  LDEis  leffe  then  the  triangle  £Z>£, 

Wherfore alfo  the  triangle GDE is leffe  thentfhe 
triangle  F  D  £.But  the  triangle  G  2)  £  is  equal  to 
the  triangle  ABC.  Wherfore  the  triangle  ABC 
is  leffe  then  the  triangle  D  E  £;  which  was  requi- 
red  to  beproued. 

But  now  let  the  angles  B  AC  and  EDF  be  leffe 
then  two  right  angles.' and  agayne  let  the  angle 
at  the  pointed  be  greater  then  the  angle  at  the 
point  Z>.Then  / fay  that  the  triagle  ABCis  grea¬ 
ter  then  the  triangle  D  E  F.Lcttbe  fame  conftru- 
ftio  be  alfo  here  that  was  in  the  two  former.  And 
forafmuch  as  the  angles  B  AC  and  £  D  £,that  is, 
the  angles  EDG  &  ED  F,  arc  leffe  then  two  right 
angle s,but  the  angles  EDG  and  GD  Katz  equal 
to  two  right  angles,  takeaway  the  angle  FDG 
which  is  common  to  them  both,  wherefore  the 
angle  remayning,  namely,  EDF  is  leffe  then  the 
angle  remayning,namely,then  GD  £T:thatis,the 
angle  H  D  iff  which  by  the  1  f .  propofition  is  e- 
quall  to  the  angle  £  D  F)  is  leffe  then  the  angle  G 
D  K.  Wherfore  the  whole  angle  G  D  H  is  leffe  then  double  to  the  angle  G  D  -KT.But  itis 
double  to  the  angleDG  Ffas  before  it  was  proued  wherfore  the  angle  GDKis  grea¬ 
ter  then  the  angle  D  G  F.  PuttheanglcDG  Lequall  to  thcangle  GDK  (by  the  23. 
propofition  Jand  produce  the  line  G  L  till  it  concurre  with  the  line  8 F alfo  produced, 
&  let  the  concurfe  be  in  the  point  L.  And  draw  a  line  from  D  to  b.Andfor  as  much  as 
the  angle  DG  Lis  equall  to  the  angle  G  D  K,  and  they  are  alternate  angles,  therefore 
the  line  G  Lisa  parallel  to  D  £(by  the  2  7. propofition ) .  Wherefore(by  this  propofiti¬ 
on  Jthe  triangles  GDE  and  L  D  £  are  equal:  bu  tthe  triangle  L  D  E  is  greater  then  the 
triangle  FD£,and  the  triangle  G  D£  is  equall  to  the  triangle  ABC.  Wherefore  the 
triangle  tABCis  greater  then  the  triangle  D  £  £:  which  was  required  to  be  proued, 

7 he  ift.T'heoreme.  T'he  fifPropofition* 

T’riangles  which  con/r/lvppon  equall  bafes,  and  in  thefelfe 
fame  parallel  lines ,are  equall  the  one  to  the  other . 

^^^Fppofe  that  thefe  triangles  A  BCand  V  E  F Jo  con  ft ft  vpon  equal  ba* 
'W0I-  jeSjtbat  ts, Vpon  B  C  and  E  Fyand  in  thefelfe  fame  parallel  lines  ytbat  is 
^S&'BF  and  A  V.Tben  I  fay  that  the  triangle  ABCis  equall  to  the  trian * 


ofEuclides  Elementes . 


FoL^p* 


Confirullton^ 


Demmffraticfi'' 


gle  A  B  C  is  equall  to  the  triangle  B)  E  F JProduce  (by  the  fecond  petition)  the 
line  AT)  on  echefede  to  the  point  es  G  and  H.  And  (by  the  3  i  .proportion)  by 
the  point  B  draiPe  Pnto  CAa  paraU 
lei  line  B  G^and(by  the  fame)  by  the  $ 
pointe  F  dra'Hoe  Pnto  IDE  a  parallel 
lineF  0  Wherfore  G  BC  A  and 
(DEF  Hare  parallelogrammes.But 
the  parallelograms  GBCA  is  (by  the 
3  6  proportion) equal  to  the paralle* 
logr  am  me  D  EF H,f or  they  conjiH 
Vpon  equall  bafesjbat  isfB  C and  E 
F,  and  are  in  the  ) elf e fame  parallel 
lines  .that  is  }BF  andG  H.  But  (by 
the  $4 .proportion)  the  triangle  A  B 
C  is  the  halfe  of  the  parallelogramme 
GBC  AJor  the  diameter  A  B  deui • 

deih  it  into  typo  e  quail  partes'-andthe  triangle  T)  EFisfby  tbefame)the  halfe 
of  the  parallelogramme  V  EF  H^for  the  diameter  FT)  deuideth  it  into  typo  e* 
quail  partes  jBut  the  h dues  ofthinges  equall  are  (by  the  7, common  fentence)e* 
quail  the  one  to  the  other. Wherfore  the  triangle  ABC  is  equall  to  the  trian* 
gle  D  EF.  Wherefore  triangles  Tphicb  con  fist  Pppon  equall  bafes,  and  in  the 
jelfe  fame  parallel  lines yare  equall  the  one  to  the  other :  TPbich  Tpas  required  to 
beproued. 

In  this  propofition  are  three  cafes. For  the  bafes  ofthe  triangles  either  haue 
one  part  common  to  them  both  or  the  bafe  ofthe  one  toucheth  the  bafe  of  the 
other  onely  in  a  point:  or  their  bafes  are  vtterly  feuereda  funder  *  And  ech  of  Ecbep  bef 
thefe  tales  may  alfo be  diuerfly5as  we  before  haue  fenein  parallelogrammes  con  /LfifomfiT 
{iff  mg  on  equall  bafes,and  being  in  the  felfe  fame  parallellines*So  that  he  which  diuerjlj . 
diligently  noteth  the  variety  that  was  there  put  touching  them!  may  alfo  eafely 
frame  thefame  varietietoechcafe  in  this  proportion*  Wherefore  I  thinke  it 
nedeles  hereto  repeat?,  the  fame  agayne-.forhowfoeuer  thebafes  beput,  or  the 
toppessthe  manner  of  confirmation  and  demonftration  here  put  by  Euclide  will 
ferue:  namely,  to  draw  parallel  lines  to  the  tides. 


..'3. 


Thre  cafes  m 
thupropofmon* 


An  addition  of  Pelitarius* 

T 0  deuide  a  triangle geuen  into  Wo  equall  partes. 

Suppofe  that  the  triangle  geuen  to  be  deuided  in  to  two 
equall  partes, be  A  B  C  .Deuide  one  of  the  fides  therof, 
name!y,5Cinto  two  equall  partes  fby  the  10.  propo¬ 
fition  jin  the  point  D.  And  draw  a  line  from  the  point  D 
to  the  point  -/Flhe  I  fay  that  the  two  triangles  A  B  D  & 
A  C  Share  Cquahwhich  is  eafy  to  proue(by  the.  3  8.  pro¬ 
portion)  if  by  the  point  A  we  drawe  vnto  the  line  B  C  a 
parallel  line  (by  the  3  1  .proportion  J^which  let  by  H  K : 
for  fo  the  triangles  AB  D  and  A  D  Cs  confiffing  vppon 
equal  bafes  B'D  &  S)C,and  being  in  the  felfe  fame  paral¬ 
lel  lines  H/Cand  B  C  are  of  neceffitie  equall.  The  felfe 

Q.i.  fame 


An  addition  ef 
I'ehtariut,to 
tleutde  a  trian¬ 
gle  into  two  e- 
yua/l partes. 


Nofe, 


An  other  addi¬ 
tion  of  Peltta- 
wtes. 


ConFiruftion. 


Vemonjlration 


7  hefirfl  Booty 

fame  thing  alfo  wil  happen  if  the  fide  B  A  be  deuided  into  two  equall  parts  in  the  point 
£,and  fo  be  drawen  a  rightline  from  the  point£,to  the  point  C.  Orifthe  fide  A  C  be 
deuided  into  two  equall  partes  in  the  point  F,andfo  be  drawen  aright  line  from  the 
point  F  to  the  point  .5:  which  is  in  like  manner  proued  by  drawing  parallel  lines  by  the 
pointes  B,-and  C,  to  the  lines  B  A  and  A  C, 

And  fo  by  this  you  may  deuide any  mangle  into  fo  many  partes  as  are  fig - 
nified  by  any  number  that  is  euenly  euen;  as  into  14, 16,32. (S^Scc. 

An  other  addition  ofpelitarius* 

From  any  point geuen  in  one  of  the  fide s  of  a  triangle ,to  draw  a  line  Which  fiat  deuide  the  trian¬ 
gle  into  tWo  equall  partes. 

Let  the  triangle  geuen  be^CD:andletthepointgeueninthefide  BC  be  A.  Itis 
required  from  the  point  A  to  draw  a  line  which  ihal  deuide  the  triangle  B  CD  into  two 
equall  partes.  Deuide  the  fide  B  Cinto  two  equall  partes  in  the  point£.  And  drawea 
right  line  from  the  point  A  to  the  point  *D.And(by  the 
3  1. proposition  J  by  the  point  E  draw  vnto  the  line  AD 
a  parallel  lineA  F:  which  letcutte  the  fide  D  C  in  the 
point  F, And  draw  a  line  from  the  point  ^4  to  the  point 
D.Then  I  fay  that  the  line  F  deuideth  the  triangle  B 

C D  into  two  equall  partes :  namely,  the  trapefium  A  B 
D  F is  equall  to  the  triangle  A  C  F. For  draw  aline  from 
E  to D,cuttine.the line  Ain  thepointC.  Nowthenit  ^ 
is  manifeft  f  by  the  3  8  .propofition)  that  the  two  trian¬ 
gles  B  E  D  and  C  £  D  are  equall  (if we  vnderftand  aline 
to  be  drawen  by  the  point  D  parallel  to  the  line  A  Cfor 
the  bafe  s  B  E  and  E  Care  equal)  .The  two  triangles  alfo  DE  F  and  A  EF  arc  f  by  the 
37.propofition)equall:fortheyconfift  vponoheand  the felfe  fame  bafe  EF,  and  are 
in  the  felfe  fame  parallel  lines  A D  and  E F.  Wherefore  taking  away  the  triangle  EFG 
which  is  como  to  the  both,the  triangl qAE  G  fhalbe  equall  to  the  triangle  DF(/  :wher 
fore  vnto  either  of  the  adde  the  trapefiu  CFG  £,and  the  triangle  ACF  fhalbe  equal  to 
the  triangle- DEC.  But  the  triangle  'DEC  is  the  halfe  part  of  the  whole  triangle  BCD 
wherefore  the  triangle^  C£is  the  halfe  part  of  the  fame  triangle  B  ^D.Wherfore  the 
refidue.namely.the  trapefium  ABF  D  is  the  other  halfe  ofthe  fame  triangle.  Where¬ 
fore  the  line  A  F  deuideth  the  whole  triangle  BCD  into  two  equall  partes :  which  was 
required  to  be  done. 

Jfi$fThe  ipJTheoreme.  T  he  tyfiPropofition. 

Equall  triangles  confining  upon  one  and  the  fame  bafe ,  and 
on  one  and  the  fame  fide :  are  alfo  in  the  felfe  fame  parallel 
lines . 


f.  P'ppofe  tbstt  thefetwo  equal!  triangles  j4.BC  andT>  BCdo  conftjl  Vp* 
^ pon  one  and  the  fame  bafe  gamely,  B  C  and  on  one  and  the  fame  fide. 
The  I  Jay  that  they  are  in  the  felfe  fame  parallel  lines.  Vra'We  a  right 
line  from  the  point  A  to  the  point  ID. Noth  I  Jay  that  A  ID  is  a  parallel  line  to 
IB  C.  For  if  not ,  then  (by  the  3  r.  propofition )  by  the  point  A  dr  all?  e  Vnto  the 
right  line  BCa  parallel  line  jEyanddraH?  a  right  line  from  the  point  E  to  the 

point 


Folio. 


of Euclides  Elementes. 

point  C.  Wherfore)  triangle  EB  Cis 
(by)  n.j propofitio)equal  to  the  triangle 
A  B  CJor  they  conji/l  vpon  one  and  the 
felfe  fame  bafe, namely  fBC^and  are  in) 
felfe  fame  parallels ,  that  is y  A  E  and  B 
C.  But  the  triangle  DBCis  (by  fuppo • 
fition )  e quail  to  the  triangle  ABC, 

VV her fore  the  triangle  DB  Cis  equal 
to  the  triangle  EBCy  the  greater  Vnto 
the  lefle.'tohicb  is  impofiible.  Where • 
fore  the  line  A  E,  is  not  a  parallel  to 
the  line  B  CtAnd  in  like  forte  may  it  be 
proued  that  no  other  line  befides  is 
a  parallel  line  to  B  Cohere  fore  AT)  is  s 

a  parallel  line  to  B  CWherfore  equall 
triangles  confifiing  vpon  one  and  the  fame  bafe,  and  on  one  and  the  fame  fide, 
are  alfo  in  the  felfe  fame  parallel  linesi'frbicb  Tvas  required  to  be  proued 

This  propofition  is  the  conuerfe  of  the  37.propofition.And  here  is  to  be  noted  This  Theorem* 
that  if  by  the  point  A.you  draw  vnto  the  line  B  C  a  parallel  line, the  lame  dial  of  •heconuerfeof 
neceffitie  either  light  vpo  the  pointD,orvndcr  it, or  aboue  it.  If  it  light  vpo  it, 
then  is  that  manifeft  which  is  required: but  ifit  light  vnder  it,then  foloweth  that 
abfurdi  tie  which  Euciide  puttech, namely,  that  the  greater  triangle  is  equall  to 
the  leffe:  which  felfe  fame  abfurditiealfo  will  follow,ifitfall  aboue  the  point  D* 

As  for  example. 

Suppofe  that  thefe  equall  triangles  A*B  Cand  T>  SC  do  confift  vppon  one  and  the 
felfe  fame  bafe  B  C,and  on  one  and  the  fame  fide.  Then  / 
fay,that  they  are  in  the  felfe  fame  parallel  lines.and  that 
a  right  line  ioyning  together  their  toppes  is  a  parallel  to 
the  bafe  B  C.Draw  a  right  line  fro  A  to  D.Nowifthisbe 
not  a  parallel  to  the  bafe  B  C,let  AS  be  a  parallel  vnto  it, 
and  let  AS  fall  without  the  line  AD,  And  produce  the 
line  B  D  till  it  concurre  with  the  line  A  E  in  the  pointe  E 
and  draw  aline  from  E  to  C.Whcrfore  the  triangle  tsf  B 
Cis  equal  to  the  triangle  EB  C:but  the  triangle  A*B  Cis 
equall  to  the  triangle  D  B  C: Wherfore  the  triangle  SBC 
is  equall  to  the  triangle  D  B  C.  Namely,the  whole  to  the 
part:  which  is  impofiible. Wherfore  the  parallel  line  fal- 
leth  not  without  the  line  A  D.  And  Euciide  hath  proued 
that  it  falleth  not  within.  Wherfore  the  line  AD  is  a.  pa¬ 
rallel  vnto  the  line  3  C.  Wherfore  equall  triangles  which 
are  on  the  felfe  fame  fide,  and  on  one  and  the'felfe  fame 
bafe,are  alfo  in  the  felfe  fame  parallel  lines:  which  was  required  to  be  proued. 

An  additionofFluffates, 

The  felfe  fame  alfo  followeth  in  parallelogrames.Forifvpon  the  bafe  AB  be  Anadditiencf 
fet  on  one  &c  the  fame  fide  thefe  equal  parallelogrames  ABCD  sc  A  B  G  E,they  riujpstes. 
fhallof  neceffitiebem  the  felfe  fame  parallel  lines.Fori  foot, but  one  of  them  is 

O.ii,  fet 


t*.  'J  f 


sirt  addition  ef 
Campanula 


Tbefirfl  Hooke 

feteyther  within  or  without,  let  the  parallelo¬ 
grams  B  F  being  equall  to  the  parallelograme  A 
BCD  be  fet  within  the  fame  parallel  lines: 
wherefore  the  fame  parallelograme  B  F  beyng  e- 
quall  to  theparallelograme  A  B  C  D  (by  the 
proportion)  (hall  alfo  be  equall  to  theother  pa- 
rallclograme  A  B  G  E  (by  the  firft  common  fen- 
tence)  For  the  parallelograme  AB  GE  is  by  fup$ 
pofition  equall  to  the  parallelogrammeAB  C  D; 
whetforc  theparallelograme  B  F  being  equall  to 
the  parallelograme  AB  G  E,the  parte  fhall  bee  e- 
qualto  the  whole, which  is  abfurde.The  fame  in- 
conuenience  alfo  will  foliowc,  if  it  fall  without* 

Wherefore  itcan  neither  fall  within  nor  withs 
out.Wherfore  equall  parallelogrames  beyng\rpon  one  and  the  felfc  fame  bafe 
and  on  one  and  the  fame  fide,are  alfo  in  the  felfe  fame  parallel  lines.  . 

/  An  addition  of  Campanus. 

If  a  right  line  deride  ttyo fide:  of  a  mangle  into  m>o  equall  partes :  'it  fall  be  efuidiffant  vntd 
the  third fide. 


Suppofe  that  there  be  a  triangle  ABC:  and  let  there  bee  a  right  ly  ne  D  E, 
which  let  deuide  the  two  fides  AB  and  B  C  into  two  equall  partes  in  the  pointes 
D  and  E  Then  I  fay  ,that  the  line  D  E  i  s  a  parallel  to  the  line  A 
C4  Drawe  thefe two  lines  AEand  D  C*Now  then  imagining 
a  line  to  bedrawn  eby  the  point  E  parallel  to  the  line  A  B,  the 
triangleBDE  fhall  (by  the  38.pro pofition)  bee  equall  to  the 
triangle  D  A  E(for  their  two  bales  AD  and  D  Bare  putto  be 
equall)  And  by  the  fame  reafon  the  triangle  B  D  E  is  equall  to 
the  triangle  C  E  D.  V  Vherforefby  the  firft  common  fen tece) 
the  triangles  E  AD  and E  C  D  are  equall,and  they  are  ere&ed 
on  one  and  the  felfe  fame  bafe,namely  ,DE,andon  one  and  the 
fame  fide. VVhereforefby  theqp^propofition)  they  are  in  the 
felfc  fame  parallel  lines,  and  the  line  which  ioyneth  together 
their  toppes  is  a  parallel  to  their  bafe.  V  Vherfore  the  lynes  DE  and  A  C  arepa- 
rallels  :  which|was  required  to  beproued* 


be^o.T heoreme.  The  ^.ofPropofition* 

Squall  triangles  confijling  vpon  equall  bafes ,  and  in  one  and 
the fame fide:  are  alfo  in  the  felfe fame  parallel  lines . 


iofe  that  thefe  equall  triangles  A  EC  and  CD  E  do  confift  Vppon  e* 
Squall  bafes  9  that  isy  Vppon  ft  C  and  CE ,  and  on  one  and  the  fame fyde , 
ti®  namely  yon  the  jide  of  A.  Then  I fay  that  they  are  in  the felfe fame  paraU 

lei 


ofEuclides  Elementes.  Fol.ji. 

Ml  lines .  Draw  by  the  firB  peticion  a 
right  line  from  the  point  A  to  the  point  e 
’  B)JS[ow  I  fay  that  A  T)  is  a  parallel  line 
to  B  E.Forifnotjhen  (by  the  y.propo* 

Jition)by  the  point  A  dray?  Vnto  the  line 
BE  a  parallel  line  A  F.  And  dralbe  a 
right  line  from  the  point  Fto  thepomte 
EyVherfore(b)  the  js, propoJitio)the 
triangle  B  A  Cis  equall  to  the  triangle 
CF E'for  they  confift  Vpon  equal  bafesy 
that  is  B  C  and  C  E,and  are  in  the  felfe 
fame  parallel  lines, namely,  B  E  and  A 
F.But  by  fuppofttion  the  triangle  A  B  ^ 

C  is  equal  to  the  triangle  CDEJ/Vher * 
fore  the  triangle  DC  Eis  equall  to  the  triangle  FCE, namely  s  the  greater  Vnto 
tbelejfe}'tobich  isimpofiible.Wberfore  A  F  is  not  a  parallel  line  to  B  Et  And 
in  lib  forte  may  Tbe proue  that  no  other  line  befides  A  D  is  a  parallell  line  to  B 
E.Wherfore  AD  is  a  parallel  lyne  to  BE  Equall  triangles  therfore  confining 
Vppon  equall  bafes^and  in  one  and  the  fame  fideiare  alfo  in  the  felfe  fame  pa* 
rallel  lines:  lubicb  "teas  required  to  be proued. 

This  propofition  istheconuerfeot:  the38,pcopo.(ition*And  inthisas  in  the 
former  propofition,ifthc  parallel  linedrawen  by  the  point  A,  Ihould  not  paffe 
by  the  poincD,  it  mud  paiTe  eyther  beneath  it,or  aboue  it. Eudide&tceth  forth 
onely  the  abfurdity  which  ihould  follow  if  it  pafle  beneath  it:  bat  the  felfe  fame 
abfurditicalfo  wdfoliow  ifit  ihould  paileaboue  it:  as  itisnothardto  lee  by  the 
gatheringthereofinche  former  propofition*  And  therefore  here  I  omitte  it* 

The  3 1 .  Theoreme.  The  4 ifPropoftion . 

If i par allelograme  &  a  triangle  haue  one  &  the  felfe fame 
bafe,  and  be  in  the  felfe  fame  parallel  lines  :  the  parallel 
grame/halbe  double  to  the  triangle. 


E  T>  A 


let  the  be  in  the  felfe  fame  parallel  lines, 
namely  fB  C&jt  E.The  I fay  ft  hat  the 
paralklograme  ABCDis  double  to  the 
triangle  B  E  CJDraief  by  the firjl peti * 
cion )  a  rygbt  line  from  the  point  e  A  to 
y  pointe  Cyyherjore(by  the  37.  propo • 

Oaf  fition 


Vppofe  that  the  paralklo  * 
grame  ABCD and  tbetri* 
angle  EBC haue  one  the 
fame  bafepiamely^  B  C,  and 


Ctnftru&M#. 


Demonfiration 
leading  to  an 
abfurditis^ 


T hh  proportion 
it  the  Conner fe 

of  the 
pofitiiu 


DemenBfitttem 


) 


(TbejirflcBooJ{C 


Jition )  the  triangle  ABC  is  equall  to 
the  triangle  EBC-.  for  they  are  Vppon 
one  and  the  ftlfe  fame  baje  &  Cy  and  in 
the felfe fame  parallell  lines  B  C  and  E 
A :  hut  the parallelogram e  A  B  CS)  is 
double  to  the  triangle  ABCfby  the  34, 
propofition  )for  the  diameter  thereof  A 
Cdeuideth  it  into  tUso  equal  part  slither 
fore  the parrallelogramme  ABCS)  is 
double  to  the  triangle  E  B  Clftherfore 
a  parallelogramme  and  a  triangle  haue 
one  and  the  felfe  fame  bafe ,  and  be  in  the  felfe  fame  parallels ,  the  parallel 
gr amefhall  be  double  to  the  triangle:  -frhicb  was  required  to  be  proued. 


Tv/o  cafet  in 
tbif  propofition. 


This  propofition  hath  two  cafes.For  thebafebcyng  one,the  triangle  may 
haue  his  toppe  withoutthc  parallelograme,  or  wichin/The  firftcafe  is  demon- 
ftrated  of  the  authorfThe  fecond  cafe  is  thus. 


Suppofe  that  there  be  a  parallelograme  AB  CD> and  let 
the  triangle  be  E  C  Neither  of  which  let  haue  oneand  the 
felfe  fame  bafe,  namely,  CDt  and  let  them  be  in  the  felfe 
fame  parallel  lines  CD  and  A  B ,  and  let  the  toppe  of  the 
triangle  £  C  ^namely, the  point  £,be  within  the  paralle¬ 
lograme  ABC  .D.Then  /  fay  that  the  parallelograme  A  2 
C  D  is  double  to  the  triangle  EC  D.  Draw  a  right  line  fro 
the  point  A  to  the  point  D.Now  forafmuch  as  the  paral¬ 
lelograme  A  B  CD  is  double  to  the  triangle  AC  D  (by  the 
34-propofition) and  the  triangle  tsfDCis  equall  to  the 
triangle  £  DC  (by  the  3  7.  propofition  J.Therfore  thepa- 
rallelogrammc  A  B  CD  is  double  to  the  triangle  £  CD; 
which  was  required  to  be  proued. 


AES 


jcorolUrj.  By  this  propofition  it  is  manifeft:  that  if  the  bafe  be  doubled,  the  triangle  c* 

refted  vppon  it  flialbe  equall  to  the  parallelogramme. 

The  felfe  fame  And  if  the  bafes  ofthe  triangle  and  of  the  parallelogramme  be  equall ,  the 

demonfirattm  felfe  fame  demonftration  will  lerue  if  you  drawc  the  diameter  oi  the  parallelo- 
Trifngk  &  tic  grame*  F°r  the  triangles  being  equal ,  the  parallelogramme  which  is  double  to 
parallelograme  the  one,  fiial  alfo  be  double  to  the  other.  And  the  triangles  muft  nedcs  be  equal! 
beSpon  tjHad  (by  the  38.propofition)for  that  their  bafes  arc  equal,and  for  that  they  are  in  the 
felfe  fame  parallel  lines. 

The  conucrfeofthis  propofition  is  thus. 

If a  parallelogramme  and  a  triangle  haue  one  And  the felfe fame  bafi,or  equall  bafes  the  one  to 
t he  at  her, and  be  deferi  bed  on  one  and  the fame  fide  ofthe  bafe  :  if  the  parallelogramme  be  double  to 
the  triangle, they  Jhalbe  in  the felfe fame  parallel  lines. 

Theconuerfe  of  For  if  they  be  notjthe  whole  fiial  be  equal  I  to  his  parte.  For  then  the  toppe 

tbit  propofition.  of  the  triangle  muftnedes  fall  either  within  the  parallel  lines  or  without.  And 

whether 


ofEuclides  Elementes .  F0L52. 

whether  of  both  foeuer  it  do,  one  and  the  felfe  fame  impoffibilitie  follow, 

if  by  the  toppe  of  the  triangle  be  draweti  vnto  the  bafe  a  parallel  line. 


An  other  conuerfe  of  the  fame  proportion* 

If  a  parallelogramme  be  the  double  of  a  triangle, being  both  Within  the felfe fame  parallel  lines  X  other  cat- 

then, are  thty  vpon  one  and  the felfe fame  bafe, or  vpon  equallbafes  .F  or  if  111  that  Cafe  their  has  uerfe  ofthe  ~ 
fes  fliould  be  vnequal,  then  might  ftraight  way  be  pro ued,  that  the  whole  ise-  famefr3?cfair>' 
quail  to  his  part: which  isimpoffible* 


A  trapefium  hamng  two  fides  onely  parallel  lines,is  ey  ther  more  then  don-  Compart  fort  of a 
ble ,  or  leffe  then  double  to  a  triangle  contayned  within  the  felfe  fame  parallel  triangle  and# 
lines, and  hauingone  and  the  felfe  fame  bafe  with  the  trapefium, or  table.!  oil  the 
double  it  cannot  be, for  then  it  fhould  be  a  parallelogramme.  A  trapefium  ha?  felfe- fame  up, 
uing  two  fides  parallels  hath  ofnccelTitic  the  one  of  them  longer  then  the  other:  and  in  the  felfe 
for  ft  they  were  equall  then  fliould  the  other  two  fides  encloiing  them  be  aifo  e-  tarallel 
quail  (by  the  35*propofition»)Ifthe  greater  fide  of  the  trapefium  be  thebafe.of  taes* 
the  triangle, then  dial  th<“  trapefium  be  leffe  then  the  double  ofthe  triangle  And 
if  the  leffe  fide  of  the  trapefiu  m  be  the  bafe  of  the  triangle  then  fhall  the  trapefi- 
um  be  greater  then  the  triangle. 


For  fnppofe  that  AB  CD  be  a  trapefium,and  let 
two  fides  thereof,  namely,  AB  and  CT>  be  parallel 
lines,and  let  the  fide  AB  be  leffe  then  the  fide  CD,  & 
produce  the  fide  AB  infinitlye  on  the  fide  B  to  the 
point  i7.  And  let  the  triangle  E  CD  haue  one  and  the 
felfe  fame  bafe  with  the  trapefium,  namely,  the  line 
CD.Then  I  fay  that  the  trapefium  AB  CD  is  leffe  the 
the  double  ofthe  triangle  £  CjD.Forput  the  line  AF 
equall  to  the  line  C  D  (  by  the  3  .propofitio)anddraw 
a  line  from  D  to  F.  Wherefore  A  CD  F  is  a  parallelo¬ 
gramme  (  by  the  3  3 .  propofition) .  Wherefore  (by 
the  3  4  propofition)it  is  double  to  the  triangle  £  CD. 
But  the  trapefium  AB  CD  is  a  part  of  the  parallelo¬ 
gramme  A  CD  F.  Wherefore  the  trapefium  AB  CD 
is  leffe  then  the  double  of  the  triangle  £CD:  which 
was  required  to  beproued. 

' 

Agayne  let  the  triangle  haue  to  his  bafe  the  fide 
A  .S.Then  I  fay  that  the  trapefium  A  2?  CD  is  trea¬ 
ter  then  the  double  of  the  triangle  tAE  2. For  from 
the  fide  C  D  cut  of  the  line  CF  equall  to  the  line  A  B 
(by  the  3 .  propofition)  And  draw  a  line  from  B  to 
F,  Wherfore  (by  the  3  3  .propofition )  <lAB  C  Fisa. 
parallelogramme.- andjrherefore  is  (by  the  344*0- 
pofition  J  double  to  the  triangle  csEEB.  Where¬ 
fore  the  trapefium  B  CD  is  more  then  thedou. 
ble  ofthe  triangle  tAEBi  which  was  required  to 
beproued. 


c  r 


When  the  grea¬ 
ter  fide  of  the 
trapefum  is  the 
bafe  of  the  tri- 
angle. 


When  the  leffe 
fide  is  the  bafe. 


•.  r- 


0.iiij, 


T 


T3 


C&njlruftton. 


ThefirttHookg 

The  u/Probleme .  'The  fyi.propofition. 

V nto  a  triangle geuenjo  make  a parallelngrame  equal ftihofe 
angle [hall  bee  quail  to  a  reBtline  angle  geuen. 

'Vppofe  that  the  triangle  geuen  beABCy  and  let  the 
retliline  angle  geuen  be  It  is  required  that  Vnto  the 

[triangle  ABC  there  be  made  a  paradelograme  equally 
‘tohoje  angle Jhal  be  equall  to  therefliline  angle  geuen, 
namely, to  the  angle  DDeuide( by  the  io ,propojitio)the 
line  B  C  into  two  equall  partes  in  the pointe  E.  And(by 
the  jirH  petition)  dra'ta  a  right  line  from  the  point  A  to 
the  point  E.And(by  the  2$.  proportion)  Vponthe  right 
Une geuen  E  C,and  to  the  point  in  it 
geuen  E}make  the  angle  C  EE equal 
to  the  angle  D.Andfby  the  5 1.  pro • 
pojition)  by  the  point  A  dr  alt)  Vnto 
the  line  E  C  a  parallel  line  A  H:  and 
let  the  line  EFcut  the  line  AH  in 
the  point  F.and(hy  thefame)hy  the 
point  C ,  dralbeyntothe  lineFFa 
parallel  line  C  G.W  her  fore  FECG 
is  a  parallelograms.  And  forajmuche 
as  BE  is  equall  to  E  Cither  fore  (by 
the  i  proportion)  the  triangle  A  .  __ 

BE  is  equall  to  the  triangle  A  EC, 
for  they  conjM  ypo  equall hafes  that 

is  BE  and  EC,  and  are  in  the  felfe fame  parallel  lines, namely,  BCandA  H 
Wherfore  the  triangle  ABC  is  double  to  the  triangle  A  E  C.And  the  parade * 
lograme  CEFGis  alfo  double to  the  triangle  A  EC:  for  they  haue  one&  the 
Jetfe  fame  baJeynamely,E  Ci  and  are  in  the  felfe  fame  parallel  lines,  that  is, EC 
and  AH.  Wherfore  the  parallelograms  BE  CG  is  equall  to  the  triangle  A  B 
C,and  hath  the  angle  C  E  F  equall  to  the  angle  geuen  D,  Wherefore  ynto  the 
triangle  geuen  AB  Cis  made  an  equall  pqrallelogr ante, namely, F EC  Gfyyhofe 
angle  C  E  Fis  equall  to  the  angle  geuen  D:  'tobicb  Tt>as  required  to  he  done4 


t>emo»pration 


D 


-  *■>  v\\  U  4V.O1 


The  conuerfe  of 
this  former  fro- 
p  option. 


orti  n 

The  conuerfe  ofthis  proportion  after  Pelitarius . 

Vnto  afarattelogramme  geuen,  to  make  a  triangle  equall,  hauyng  an  angle  equall  to  a  reflilint 
angle  geuen.  1  ■  ' 

Suppofe  that  the  parallelograrae  geuen  be  A  B  C7),  and  let  the  angle  geuen  be  Ea 
Itis  required  vnto  the parallelogtaine  *4 'BC  T>  to  make  a  triangle  equall  hauyng  an 

angle 


ofEucIidesElementes. 


Fol.%. 


A  '  B 


m 


angle  equal  to  the  angle  £.Vpon 
the line. CD  and  to  the  pointe  in 
it  C,defcribe  (by  the  2  3 .  propo- 
pofition)  an  angle  equall  to  the 
angle  Ei  which  let  be  DCF:  aud 
let  the  line  C  F  cut  the  line  cA'  B 
being  produced,  in  the  point  F: 
and  produce  the  line  C I)  (which 
is  paralfel’ to  the  line  A  F)  to  the 
point  G.And  put  the  line  DGe- 
quall  to  the  line  C  D  and  draw  a 
line  from  F  to  Cj,  Then  / fay  that 
the  triangle  C  F  G  is  equal  to  the 
parallelograme  ABCD.  For  for¬ 
afmuch  as(by  the  38.  propofition )  the  whole  triangle  CF  G  is  double  to  the  triangle 
CDF .  Aud(  by  the  4 1  .propofition)the  parallelograme  ABC  D  isdouble  to  the  fame 
triangle  C  D  Fi  therfore  the  parallelograme  A  BCD  and  the  triangle  CFG  are  equall 
the  one  to  the  other:  which  was  required  to  be  done. 


n 


The  yiftbeoreme.  The  4.3. Tropofition . 

fn  euery parallelograme fbe fupplementes  ofthoje paralklo* 
grammes  which  are  about;  the  diameter  ?are  efuaUthe  one  to 
the  other.  ? 

Vppofe that  ABC  (Dbea parallelograme, and  let  the  diameter  then 
of  be  A  C:  and  abou  t  the  diameter  A  Clet  thefeparadelogrames  EM 
and  O  Fconfift:  and  let  the  fupplementes  be  B  Kjmd  KJD.  Then  I 
fay  that  the/upplement  B  Kjs  equall  to  the  fupplement  K,!D.  For 
fora/much  as  ABCD  is  a  parallelograme 
and  the  diameter  therofis  A  C}  therfore  & 

( by  the  3  4. propofition )  the  triangle  A 
B  C  is  equall  to  the  triangle  A£>C,  A * 
gayne  forafmuch  as  A  EKJA  is  a  pa* 
ratlelograme3and  the  diameter  therofis 
A  therfore  (by  the  fame)  the  trian* 
gle  A  E  Kjs  equall  to  the  triangle  A  M 
K.And  by  the  fame  reafon  alfo  the  tri¬ 
angle  KJFC  is  equall  to  the  triangle  K^ 

G  C,  And  forafmuch  as  the  triangle  A  E 
Kjs  equall  to  the  triangle  A  HK+and 
the  triangle  i^FC  to  the  triangle  K  G 
C,  therfore  the  triangles  A  E  Kjtnd  K, 

G  C  are  equall  to  the  triangles  A  H K 

and  K.EC:  and  the  whole  triangle  A  B  Cis  equall  to  the  whole  triangle  A  !D 

<Pf  C 


v<-  Thefir/i^ooke 


H  o  tp  parallclo- 
grammes  are 
fajdeto  conffle 
about  a  dia¬ 
meter. 


C:  Tbherfore  the  refidue, namely  ,the  fupplement  ©  KJs  (by  the  common  fen* 
tence)  e  quail  to  the  refiduejiamelyjo  the fupplement  I(J) .  Wherefore  in  e* 
uery parallelogr amme ,  thefupplementes  of tbofe parallelogrammes  'tobicbe  are 
about  the  diameter,  are  equall  the  one  to  the  other :  Tbhiche  Tt>as  required  to  be 
proued , 

Thofe  paralldcgrames  are  fayde  to  confift  about  a  diameter  which  haue  to 
their  diameters  partofthc  diameter  ofthe  whole  and  great  parallelograme,  as 
in  the  example  of  Euclidc,  Andfuchparallclb- 
grames  whichhaue  not  to  their  diameters  part 
ofthe  diameter  ofthe  greater  parallelograme, 
are  fay  de  not  to  confift  about  the  diameter,^  or . 
the  the  diameter  ofthe  greater  parallelograme 
cutteth  the  fide  outlie  Idle  cocayned  wy  thin  it. 

As  in  the  parallelogramme  A  i?,thc  diameter  <7Z>, 
cutteth  the  fide  EH  of  the  parallelogramme  CE. 

Wherefore  the  parallelogramme  CEis  not  about  j 
one  and  the  felfe  fame  diameter  with  the  parallelo- 
grammeC®.  > 


H 


C 


r~ 

■  yAf:  y  . 

t  • ;  > 

* 

E  .  , 

3 


Supplementes  or  Complementes  are  thofe  figures,  which  with  the  two  pa- 
Sctmpkm!nS.  lallelogrpmmcs accomplifinhewholeparan-clogramme.  Although  Peiitarius 
for  diftmttion  fakeputteth  a  difference  betwene  Supplementes  and  Comple¬ 
mented, The  parallelogram  mes  aboutthc  diameter  he  'calieth  Complementes, 
the  other  two  figures  he  calieth  Supplementes* 


Three  cafes  in 
thisTheorcmc. 


7  he  firft  cafe. 


...  This  theorerae  hath  three  cafes  onely*  Fortheparallelogrammes  which 
confift  about  the  diameter,  eythcr  touch  the  one  the  other  in  a  point:  or  by  accc 
ray  ne  parte  of  the  diameter  are  feuered  the  one  from  the  other:  or  els  they  cutte 
the  one  the  other .  For  the  firft  cafe  is  the  example  ofEuclidebefore  fct.  The  fc- 
cond  cafe  is  thus.  •  '  •  :  ’ 


The  fecond  cafe. 


Suppofe  that  AB  be  a  parallelograme? 
whofe  diameter  let  be  CD  :  andabpute  tfie 
fame  diameter  let  thefe  parallelogra’mmes  C 
KandD  L  confift:  and  betwene  the  let  there, 
be  acertavne  part  of  the  diameter,  namely, . 

L  K.Then  I  fay  that  the  two  fupplementcs  A 
O  L  ICE  &  IS  F  KL  //are  equall.  For  we  may 
as  beforefby  the  3  4.propofition  jproue  that 
the  triangle  AC D,  is  eq uall  to  the  triangle 
B  CD, and  the  triangle  E  C  K  to  the  triangle 
K  CF,  and  alfo  the  triangle  D  G  L  to  the  tri-  c 
angle  D  H  Lt  Wherfore  the  refidue, namely, 
the  fiuefided  figure  AG L  KE  is  equall  to 
the  refidue, namely,  to  the  flue  fided  figure  B 
F  K  L  Hi  that  is,  the  one  fupplement  to  the 
other  .which  was  required  to  be  proued. 


The th>rd cafe.  But  now  fuppofe  &A B  to  be  a  paralleIogramme,and  let  the  diameter  thereof  be 

CD:  and  let  the  one  of  the  parallelogrammes  about  it  be  E  C  F  L9  and  let  the  other  be 

s?  :*  DGKHi 


of  Euclides  Elementes. 


FoL 54. 


A 


Jp 


DGKH,o£which  let  the  one  cut  the  other. 

Then  I  fay  that  the  fupplementes  F  G  and  EH 
areequall.For  forafmuchas  the  whole  trian¬ 
gle  DGK  is  equal  to  the  whole  triangle  DHK 
fbythe  jq.propofition  J, and  part  alfo  of  the 
onemarndy,  the  triangle  KL  Mis  equal  to 
partoftheother.namely^to  the  triangle  KL 
iV(by  the  fame),  forLKis  aparallelograme  i 
therefore  the  refidue,naraelyJthe  Trapefium 
©TA^Ais  equall  to  the  refidue,  namely,  to 
the  trapefiu  D  L  MG*  but  the  triangle  ADC 
is  eq  ual  to  the  triangle  BCD,  and  in  the  pa- 
railelograme  E  F,  the  triangle  F  CL  is  equal! 
to  the  triangle  £C£,and  the  trapefium  D  G 
MLisf  asit  hath  bene  proued)  equal!  to  the 
trapefium DHNE.  Wherefore  the  refid ue, 

namely,  the  quadrilater  figure  G  F  is  equall  to  the  refidueuiamely,  to  the  quadrilater 
figure  E Hjhxt  is, the  one  fupplement  to  the  other:which  was  required  to  be  proued. 


This  is  to  be  noted  that  in  ech  ofthofe  three  cafes  it  may  fo  happen, that  the 
parallelogrammes  aboute  the  diameter  {hall  not  haue  one  angle  common  wy  th 
the  whole  parallelogramme,as  they  haue, in  the  former  figures.But  yet  though 
they  haue  not, the  felfe  fame  demonftration  wil  feme,  as  it  is  play  ne  to  fee  in  the 
figures  here  vndcrneathput*Foralwayes,  if  from  thinges  equall  be  taken  a^ay 
thinges  equall,thc  refidue  fhalbe  equall. 


/ 


This  propofition  P  elitarius  calleth  Gnomicall,  and  mifticall,for  that  of  it 
(faythhe)  fpring  infinite  dcmonftrations,andvfes  in  geometry.  And  he  putteth 
the  conuerle  thereof  after  this  manner*  andmifttcal. 

V  ;  *  "  -  ’  '( i  <  '  *  *  'V 

If  a  parallelogramme  be  derided  into  Wo  equall  fupplementes ,and  into  Wo  complements  What-  The  conuerfe  of 
foeuer:  the  diameter  of  the  Wo  complementes  JhaH  be  fit  direttly ,  and  make  one  diameter  of  the  this  propofition. 
Vehole  parallelogramme. 

Here  is  to  be  noted  as  1  before .  .0  lied  thatpeli  tarius  for  diftindion 
fake  putteth  a  diflerencebetwene  fupplementes  and  complementes  /which  diffe- 
rencejfor  that  I  haue  before  declared,  I  fhallnot  needeheretorepeteagayne. 

Suppofe  that'  there  be  a  parallelogramme  A  BCD,  whofe  two  equall  fupplements 
let  be  A  £  F  (./  and  F  HD  K,and  let  the  two  complementes  thereof  be  GF  C  K  and  £5 
££T: whofe  diameters  let  be  CF  and  FA.Then  I  fay  that  CFB  is  one  right  line,  and  is 
thediameter  of  the  whole  parallelogram  me  A  B  CD\  forifit  be  not,  then  is  there  an 

F-ij*  other 


/ 


GtnRruttisn. 


Tbefirft'Bookp 


other  diameter  of  the  whole  parrallelogrammewhich  let 
be  C  L  B  being  drawen  vnder  the  diameters  C  F  and  F  B, 
and  cutting  the  the  in  the  point  L,And( by  the 

1 1  .propolition  )  by  the  point  L,draw  vnto  the  line  a A  C 
a  parallel  line  M  L  N.  And  fo  are  there  in  the  whole  paral¬ 
lelogramme  A'B CD  two fupplements  AMGL and  L H 
N  D, which  by  this  propolition  lhalbeequall  the  one  to 
the  other.  For  that  they  are  about  the  diameter  C  L  B. 
Butthefupplementex4r£.F(7is  fbyfuppolition)  equall 
to  the  fupplcment  F  H  D  K:and forafmuch  as  FHD  Kis 
greater  then  LHDN,AEFG  alfo  dial  be  greater  then  A 
MGL,nameIy,the  part  greater  then  the  whole:  which  is 
impolfible.  And  by  the  fame  reafo  may  it  be  proued,that 
the  diameter  cannot  be  drawen  aboue  the  diameters  C  F 


-t 


and  F  B.  Wherefore  C  F  B  is  one  diameter  of  the  whole  parallelogramme  tABCD: 
which  was  required  to  be  proued. 


H*The  rz.  Trobleme .  The 44. Tropofition . 

Vppon  a  right  line geuen  Jo  apply  e  a  paraUelograme  equall 
to  a  triangle geuen,  and contayningan  angle  equall  to  a  rev * 
tiline  angle geuenm 


7M  V pp°ft  that  the  right  litiegeuen  he  A  !E,and  let  the  triangle  geuen 
yjibe  Cyand  let  the  reSliline  angle  geuen  be  (D.Itis  required  Vpon  the 
I  right  line  geuen  AByto  apply  e  a  parallelogramme  equal  to  the  trian* 
gle  geuen  Cy  and  contayning  an  angle  equall  to  therectiline  angle ge* 
ue  (D,T)efcribe(by  t he  4 fypropo/i tion)y 
paraUelograme  EG EF equall  to  the  tri* 
angle  Cy  and  hailing  the  angle  <BGFe* 
quaU  to  the  angle  (D.And  vnto  the  line  E 
<B  ioyne  the  line  A  E  in  fuch  fort  that  they 
make  both  one  right  line.  Jnd  extend  the 
line  EG  beyond  the  point  G  to  thepoynte 
HtAnd(by  the  ]ipropo/ition)by  the  point 
A  draive  to  either  ofthefe  lines  E  G  and 
EE  a  paraUellme  A  H.  And  (by  the  fir  ft 
peticion)draf»  a  right  line  from  the  point 
Hto  the  point  E.And  forafmuch  as  vpon 

the  parallel  lines  A  Hand  E  Efalleth  a  certayne  right  line  H F,  there  fore(by 
the  29  propofition)the  angles  A  HEandHEE>are  equaUto  two  rightangles: 
therefore  the  angles  E  H  G  andGFE  are  lejje  then  th>o  right  angles  :  but  if 
ypon  tivo  right  lines  fall  a  right  line  making  the  inward  angles  on  one  and  the 

fame 


I 


ofEuclides  Ekmentes.  FoL $£* 

fame fide  lejfe  then  tfro  right  angle  s^thofi  right  Itnes  being  infinitly  produced 

JhaU  at  the  length  mete  on  that  fide  in  which  are  the  angles  lejfe  then  tfro  right 

angtes(by  the  5,  petition).  VVherfore  the  lines  HBand  FE  being  infinitly 

produced  "frill at  the  length  mete, Let  them  be  produced^  let  them  mete  in  the 

point  K^And  (by  the  3 1  propofition)  by  the point  KJlrafr  to  either  of  thefe  lines 

EA  and  F  Ha  parallel  line  K^LfAnd  (by  the  2. petition)  extend  the  lines  H 

A  and  GB  till  they  cocurre  "frith  the  line  H\L  in  the  pointes  L  and  M. Where*  Dem»njtr*tio» 

fore  H  L  EJFis  a  parallelogramme yand  the  diameter  thereof  is  H  Kg  and  a» 

bout  the  diameter  HKjre  the  parallelogrammes  A  G  and  M Ey  and  the  flip* 

plementes  areLB  and  BFifrhere fore  (by  the  4  3 ,  propofition)  the  (up  plem  en  t 

LB  is  equall  to  thefupplement  BFibut  by  confiruttton  the  parallelograme  BF 

is  e  quail  to  the  triangle  Cifrherefore  aljo  the  parallelogramme  L  Bis  e  quail  to 

the  triangle  C,And forafmuch  as  the  line  F His  a  parallel  to  the  line  Ly  and 

Vpon  them  lighteth  the  line  G  Mother e fore  (by  the  27.  propofition)  the  angle 

FG  Bis  equall  to  the  angle  B  ML, But  the  angle  FG  B is  equal 1  to  the  angle 

!Dytherfore  the  angle  B  ML  is  equal  to  the  angle  D.  Wherfore  Vpo  the  right 

linegeuen  A  B  is  applied  the  par rallelogr ante  L  B, equal  to  the  triangle geuen 

Qand  contayning  the  angle  B  ML  equal  to  the  reFlilme  angle geuen  iD:  frhicb 

fras  required  to  be  done* 

Applications  of 

Applications  oflpaces  or  figures  to  lines  with  exceffes  or  wantes  is  (fayth  ^ff  soffalTs 
Eudemus)  an  auncient  muention  of  Pithagoras.  a»au»dent  inn 

Mention  of  Pi- 

When  the  fpace  or  figure  is  ioyned  to  the  whole  line, the  is  the  figure  fayd  it 

to  be  applied  to  thcline.But  if  the  lengthof  the  fpace  be  longer  then  the  line, the  fajde  to  be  ap- 
it  is  fay  de  to  exceede:  and  ifthe  length  of  the  figure  be  fhorter  then  the  hne,fo  r^dtoaiine. 
that  part  of  the  line  remay  neth  without  the  figure  deferibed,  then  is  it  fayde 
to  vant. 

In  this  probleme  arc  three  thinges  geuen*A  right  line  to  which  the  applica-  ffZfJthss 
tion  is  made, which  here  muftbe  the  onefideofthe  parallelogramme  applied,  A  propofition , 
triangle  whereunto  the  parallelogramme  applied  muft  bee  equall :  and  an  angle 
wheruntothe  angle  of  the  parallelograme  applied  mull:  be  equally  And  if  the  an¬ 
gle  geuen  be  a  rightangle,the  fhal  the  parallelograme  applied  be  either  a  fquarc9 
or  a  figure  on  the  one  fide  longer*  But  if  the  angle  geuen  bean  obtufc  or  an 
acute  angle9then  flia.ll  the  parallelograme  appliedbe  a  Rhombus  or  diamond  fi- 
gurejor  els  a  Rhomboides  or  diamondlike  figure* 


The  Conuerfe  of  this  propofition  after  Peli  tarius* 

Vpon  a  right  line geuen ,to  applie  vnto  a  parallelograme  geuen  an  equall  triangle  hauyng  an  an* 
gle  equall  to  an  angle  geuen.  T^*  conuerfe  of 

tba  propofition. 


Suppofe  that  the  right  line  geuen  be  aA  J?,and  let  theparallelograme  geuen  be  C 
D  B  F*and  let  the  angle  geuen  be  G.  It  is  required  vpon  the  line  A  B  to  deferibe  a  tria- 
gleequall  to  the  parallelograme  CD  E F, hauing  an  angle  equall  to  the  angle  G.  Drawe 
the  diameter  CF&produceCD  beyond  the  pointD  to  the  point  H.  And  put  the  line 

P.iij,  D  H 


ThefirflUoohg 


Cong/uBion„ 


D£/ equall  to  the  lin eCD.  And 
draw  a  line  from  F  to  H.  Now  the 
(by  the  qi.prqpofition)  thetria- 
gle  C  H F  is  equall  to  the  paralle- 
lograine  CD  E F ^nd(by  this 
propofition)vppon  the  line  B 
defcribe  a  parallelograme  ABKL 
equall  to  the  triangle  CHi^ha- 
uing  the  angle  AUL  equal  to  the 
angle  geuen  G;  and  produce  the 
line  B  L  beyonde  the  pointe  L  to 

the  point  M.And  put  the  line  LM  equall  to  the  line  BL,and  draw  aline  from  A  to  Mv 
Then  I  fay-that  vpon  the  line  AB  is  defcribed  the  triangle  A  B  M,which  is  fuch  a  trian¬ 
gle  as  is  required.  For  (by  the  41  .propofition)the  triangle  A  BM  is  equal  to  the  paral¬ 
lelogramme  AB  K  L(  for  that  they  are  betwene  two  parallel  lines  B  M  and  A  K,  &  the 
bafe  of  the  triangle  is  double  to  the  bafe  of  the  parallelogramme)  :but  A  3  K  L  is  by 
conftrudion  equall  to  the  triangle  C  H F:and  the  triangle  CHFis  equall  to  the  pa¬ 
rallelograme  CD  E  F.Wherfpre  (by  the  firil.cpmmon  fentence)  the  triangle  ABM.  is 
equall  to  the  parallelograme  geuen  CD£  isandhath  his  angle  A  BM  equal  to  the  an¬ 
gle  geuen  C:  which  was  required  to  be  done.  *'■ 


vv 


-  .  i  *  •  s  v?  •  *, 

The  ifTrobleme,  The  ^Tropofition* 

r  ‘  '  1  u)  ri  ‘  '  •  'A ‘  ' 

l  Ky  r.  j  v  . 

To  defcribe  a  parallelograme  equal  to  any  reBiline figure  ge* 
uen^andcontayning  an  angle  equall  to  a  reUiline  angle  gene. 


i.  L’a 


Demon  fir  at  ion' 


Vppofe  that  the  re  Biline  figure  geuen  he  3  C(D,and  let  the  reBilim 

angle getie  he  E,It  isrequired  to  de/crihe  a  parallelograme  equall  to  the 
reBilme  figure  geuen  SiftCD^and  contayning  an  angle  equal  to  the  re* 
Bilim  angle  geuen  E.(Dra'to(by  thefirfi:  peticioma  right  line  fro  the  point 0  to 
the  point  n.And  (by  the  y2,propofition)vnto  the  triangle  Aft  l D  defcribe  an  e* 
quail  parallelograme  F  Mfiauing  his  angle  Fjf^FI  equall  to  the  angle  E.  And 
(by  the  4dtofthefirfi)  bpo  the  right 
line  G  H apply  the  parallelogramme 
G  A i equal  to  the  triangle  VBCfia* 
uing  his  angle  GUM  equall  to  the 
angle  E.  And for aj much  as  eyther  of 
thofe  angles  H  EfF and  G  H  M  is 
equall  to  the  angle  E :  therefore  the 
angle  MEfFis  equall  to  the  angle 
G  TIM:  put  the  angle  EfM  G  com * 
mon  to  them  both  fibber  fore  the  an * 
oles  F  KJFd and KJAG  are  equall 
to  the  angles  EfH.  G  and  GUM. 
but  the  angles  F  E^M  and  EfMG 
are  (by  the  ~g,propofition)  equall  to 
tn?Q  right  angles  yEherf ore  the  angles  KJA  G  and  G  MM  are  equall  to  two 

right 


ofEudtdes  Elementes.  FoL*>6 . 

right  angles.lSlow  then  Vnto  a  right  line  G  H,and  to  a  point  in  the  fame  H^are 
drathen  two  right  lines  Kjtdand  HM not  both  on  one  and  the  famejidey  ma <* 
king  the fide  angles  equall  to  ttvo  right  angles, W  her  fore  (by  the  i^propofiti* 
on)  the  lines  K^H  and  HM  make  dire  Elly  one  right  line.  And  forajnmch  asV* 
pon  the  parallel  lines  J\M  and  F  Gfalleth  the  right  line  H  Gy  therefore  the 
alternate  aagles  M  HG  and  H  G  Fare  (by  the  2<y  propofition  fequali  the  one 
to  the  other :  put  the  angle  H G  L  common  to  them  both^Vherfore  the  angles 
MHG  and  H G  Fare  equall to  the  angles  HG  F  andHG  L  'But  the  angles 
MHGejr  HGL  are  equall  to  ttvo  right  angles  (by  ey  29^ propofition), VF her* 
fore  alfo  the  angles  H  G  Fand  HG  L  ire  equall  to  tlvo  right  angles.  VVber* 
fore  (by  the  \4,propofitiori)  the  lines  FG  and  G  L  make  dire  Elly  one  right  line. 
And  f ora/much  as  the  line  E\Fis  (by  the  24.fr  opofition)  equal  to  the  lyne  H  Gs 
and  ft  is  alfo  parallel  Vnto  it:  and  the  line  HGisfby  the  fame)  equall  to  the  line 
MLjtherfore  (by  the  first  common fentence)  the  line  F  Kjs  equall  to  the  lyne 
MLyand  alfo  a  parallel  Vnto  it  (by  the  3  o,propofition),  But  the  right  lynes 
M and  F L  ioyne  them  together  yVherf ore  (by  the  $  $  ,propo(ition)  the  lines  E( 
M  andF  L  are  equall  the  on  to  the  other  and  parallel  Itnes.VVberfore  IfiFLM 
is  a  parallelograme.And  forafmuch  as  the  triangle  A  BID  is  equal  to  the  paraU 
lelogrameF  H}and  the  triangle  DBCto  the  parallelogramme  G  M ;  therfore 
the  whole  reEliline  figure  ABCDis  equall  to  the  tvhole  parallelograme  Kfih 
M.  VF her  fore  to  the  reEliline  figure  geuen  ABCDis  made  an  equall  parallel 
grame  KJE LM tyhoje  angle  F  KJM is  equal  to  the  angle  geuen jaamelyfo  E: 
iv fitch  'tv as  required  to  be  done. 

The  rc&ilinc  figure  geue  is  in  the  example  of  Euclide  is  aparallelograme.But 
if  the  re&ilinc  figure  be  of  many  fidcs,as  of  5.5. or  mo,themuftyo«  refoluc  the 
figure  into  his  triangles,as  hath  bene  before  taught  in  the  3 2 ♦  propofition.  And 
the  apply  a  parallelograme  equal  to  eucry  triangle  vpon  a  linegcue.as  before  in 
the  example  of  the  author.  And  the  fame  kind  of  reafoningwil  feme  that  was  be 
fore, only  by  reafoofthe  multitude  oftriangles,you  ihall  haue  neede  ofokener 
repeticio  ofchez^.and  i4*propofitiostoprouethatthebafes  ofal  the  parallel o« 
grames  made  equall  to  all  the  triangles  make  one  right  line,  and  (c  alfo  of  the 
toppes  ofthefaidparallclogramesJPelitarius  addeth  vnto  this  propofition  this 
Pro'oleme  following* 

T vne  quail  reEliline fuperficieces  beyng  geuen ,to find  out  the  excejfe  of  the  greater  aboue  the  lejfe. 

Suppofe  that 
there  be  two  vne- 
quall  re£filine  fu- 
perficieces  A8cB 
ofwhichlet^be 
thegreater.  It  is 
required  to  finde 
out  the  excefle  of 
the  (uperficies  A 
aboue  the  fuper- 
ficieces  B  .  De- 
feribefby  the  44. 

P.iiij.  pro- 


addition  of 
PelitarsM. 


'ThefifUHooke v 

propofition)the  parallelograme  CDE  F  equall  to  the  rediline  figure  -^,contayning  a 
right  angle. And  produce  the  line  CD  beyond  the  point  'Z>  to  the  point  (j  :  &  put  the 
line  T>  Cj  equall  to  the  line  C  D.  Andagaine  (by  the  44.propofition )  vpon  the  line  D  G 
dcfcribe  the  parallelograme  DG  H  X equall  to  the  rediline  figure  “2?,  and  hauyng  the 
angle  D  G  Ka  right  angle.  And  produce  the  line  K  H  beyond  the  point  H,vntiliit  cutte 
the.Iine  C  £  in  the  point  L.Then  I  fay  that  H  L  E.F,  is  the  excefie  of  the  rediline  figure 
c/Zaboue  the  rediline figure2?,Fornrft. that CGKL  isaparallelogrammeit  is  mani- 
feft, neither  nedeth  it  to  be  demonftrated.  And  forafmuch  as  the  lines  C'D  and  *D  G  are 
by  fuppofition  equal  and  either  of  them  isaparallel  to  K  L,therfore(by  the  3  G  propo- 
fitionjthe  two  parallelogrames  CH and  D  /fare  equall.  And  forafmuch  as  D  K  is  fup- 
pofed  to  be  equall  to  the  rediline  figure  £,C//alfolh  all  bo  equall  to  the  fame  rediline 
figure  rB. Wherfore  forafmuch  as  the  whole  parallelograme  CFis  equali  to  the  rediline 
figure  a^andL  F  is  the  exccffe  of  CF  aboue  D  L  or  D  K,it  followeth  that  L  F  is  the  ex- 
celfe  of  the  rediline  figure  aboue  the  rediline  figured  :  whiche  was  required  to  be 
done. 


An  other  more  redy  'fray, 

i 

Let  the  parallelograme  C  DEF  remayne  equall  to  the  rediline  figure  A,  &  produce 
the  line  C  £>  beyond  the  point  d  to  the  pointe  G  .  And  vpon  the  line  D  G  deferibe  the 
parallelogrameDG  HKequall  to  the  rediline  figure  B.  And  produce  the  lines  EC  & 
H  K  beyond  the  points  C  and  K  till  they  concurre  in  thepoiut  L.  And  by  the  pointe  D 
draw  the  diame¬ 
ter  LDM,  which 
let  cutte  the  line 
HGbcyng  pro¬ 
duced  beyonde  c 
the  pointe  G  in  ^ 
the  point  M,  & 
by  the  pointe  M 
drawc  vnto  the  E 


L  ] 

sc 

H 

. . . 

D  - - - 

• 

- - » 

"  • 

- - - ? - 

V J  '  A  ' 

J 

M 

lincHLaparal-  * 

fel  M  N  cuttyng  the  line  E  L  in  the  pointe  N :  and  by  that  meanes  is  H  L  M  N  a  paralle* 
lograme.Then  /  fay  that  N  F  is  the  eveefie  of  the  rediline  figure  aboue  the  rediline 
figure  £.For  forafmuch  as  the  parallelograme  H  D  is  equall  to  the  rediline  figure  /?,  & 
the  fupplementes H  D and D  N  ar  (by  the  43 .propofition )  equall :  therfore D  N  alfo 
is  equall  to  the  rediline  figure  B,which  rediline  figure  D  N  being  taken  away  fro  the 
parallelograme  C  F  (which  is  fuppofedto  be  equall  to  the  rediline  figure  A)  the  refi- 
due  N  F  mall  be  the  ezcelfe  of  the  rediline,  figure  A  aboue  the  rediline  figure  £:  which 
was  required  to  be  done. 


The  \\fProbleme \  The  45. Tropojition . 


Vppon  a  right  line geuen  Jo  deferibe  a  fquare. 

3  ^at  right  linegeuen  be  A  BJt  is  required  Vpo  the  right 
*  line  A  Bjo  deferibe  a  fquare. Vpon  the  right  line  A  (8,  and  from  a 
point  in  itgeuenynamely y  A9rayfe  Vp  (by  the  n  propojitton)  a  per # 
pen  diculer  line  A  C.  And  (by  the  $  propofition)  Vnto  A  B  put  an  e* 
quallline  AT).  And  (by  they.propojitm)  by  the  point  T)  dratve  Vnto  AT  a 
parallel  line  D  E.  And  (by  the  fame)  by  the  point  B  dr  awe  Vnto  A  V  a  parallel 

line 


Fol.ij. 


: ;  siolm: 

.  iH, 


a ! ;  t> f 


h::n  r:- 
■  5S2SJ 


£ 


of Euclides  Elementes • 

line  SB,  Wherefore  AD  E  B  is  a  pa* 
raUelogrammeVVberefore  the  line  A  3$ 
is  e  quail  to  the  line  D  Ey  and  the  line  A 
D  to  the  line  B  E:  but  the  line  ABise* 
quail  to  the  line  A  Dwherefore  thefefdtP 
"User  lines  B  AyA  DfD  E,E  B,  are  equall 
the  one  to  jthe  other.  Vlfhsrefore  the  pa* 
rallelogramme  ADEBconpfteth  of  e* 
quail  fides .  I  fay  alfo  that  it  is  re&angle. 

For  forafmuch  as  Vpon  the  parallel  lines  A 
B and  DEfa Ueth  a  right  line  AD:thtre * 
foref  by  the  i  ypropofition)  the  angle. :  B 

.AD  and  A  t)  Eare  equal  to  two  right  anglesihut  the  angle  BA  3)  is  a  right 
angkyV her fore  the  angle  AD  E  alfo  is  a  right  angle ,  But  in  parallelogrames 
the  fides  and  angles  which  are  oppojite  are  e  quail  the  one  to  the  other(by  the  $  4 
proportion).  Wherefore  the  two  oppojite  angles  A  B  Eand  BED  are  ech  of 
them  a  right  angle.  Where  fore  the  parallelograme  A  B  ED  is  reBangleiejr  it 
is  alfo  pr  oued  that  it  is  equilater .  VVherfore  it  is  a  (quart it  is  defer  ibed  Vpa 
on  the  right  tmegemnA  B\  which  Was  required  to  be  done # 


3 


tv  ;  'y.\  i. 


£ 


ftration-i 

ere&ed  the  perpendicular  line  CA  vpon  the  line  A  2?,and 
,put  the  line  A  E  equall  to  the  line  A  B  :  then  open  your 
compalfeto  the  wydth  of  the  line  ABorAE,tk  fetone 
foote  thcreofin  the  point  £,and  deferibe  a  peece  of  the 
circumference  of  a  circle:  and  againe  make  the  centre  the 
point  2?,and  deferibe  alfo  a  piece  of  the  circumference  of 
a  circle-jUamelydn  fuch  fort  that  the  peece  of  the  circum- 
ferece  of  the  one  may  cut  the  peece  of  the  circumference 
of  the  other,as  in  the  point  D :  and  from  the  point  of  the 
interfei5tion,draw  vnto  the  points  E  &  B  right  lines:  Sc  fo 
ihaibe  deferibed  a  fquare,As  in  this  figure  here  put,wher-  a  ^ 

in  / haue  not  drawen  the  lines  E  D  and  Z>  £,that  the  pee-  ** 

ces  of  the  circumference  cutting  the  one  the  other  might  the  plainlier  be  fene. 


To  defer i be  * 
Mire  mecket- 
ntcuUj, 


An  addition  ofProclus. 


If  the  lines  vpon  Vchich  the fquares 

Suppofe  that  thefe  right  lines 
A B and  CD  be  equall,  &vpon 
the  line  AB  deferibe  a  fquare  A 
BEG  :and  vpon  the  line  CD  de¬ 
feribe  a  fquare  CD  HF.  Then  I 
fay  that  the  two  fquares  ABE 
GScCDHF are  equal*  For  draw 
thefe  right  lines  C2?and  HD. 
And  forasmuch  as  the  right 
Iines./45andCDare  equall, & 
thelinesoA  G and HC  are  alfo 
equall,and  they  contayne  eqaul 


be  deferibed  be  equall, the /quotes  alfo  are  equall. 


angles 


The  eo'tuerfe 
fhertf* 


Conflruttiott, 


Thefir/ffiooke 

aogles,i|amely,right  angles  (by  the  definition  of  a  fqiiafe)thercforefby  the  4.  ptopo* 
fition  Jthe  bafe  B  G  is  equall  to  the  bafe//2).And  the  triangle  ABG  is  equal!  tothe 
triangle  C  D  //.Wherefore  the  doubles  of  the  faide  ttiaiiglcs  are  equall.  Wherefore 
the  fquare  A  E  is  cquall  to  the  fquare  CFi which  was  requited  to  be  proued. 

■■■  .  :  '  • '  4  .  ■  .  ■.  n ns 5 ’H-,  1  o;i  v'.. 

The  conuerfc  thereof  is  thus*.  \ 

If  the  fquares  be  equall:  the  lines  alfo  vppon  Vphicb  they  are  defeated  are  equall. 


Suppofe  that  there  be  two  cquall  fquares  A  F  and  C  Cdefbribed  vpon  the  lines  A 
B  Sc  B  (f.  The  I  fay,that  the  lines  A  B  andP  C  are  cquall.Put  the  line  *4£dire&ly  to  the 
line  2?C,that  they  both  make  on  right  line,  And 
forafmuch  as  the  angles  are  right  angles,ther- 
fore  alfo(by  the  i4.propofition)the  right  line 
FB  is  fet  dire&Iy  to  the  right  line  B  G •  Dr  awe 
thefe  right  lines  FC^  AG,AF,  and  C  G .Now  for 
afmuch  as  the  fquare  AF  is  cq ual  to  the  fquare 
CC,the  triangle  alfo  A  F B  (halbe  equall  to  the' 
triangle  CB  G:  put  the  triangle  BCF  comon  to 
them  both.  Wherfore  the  whole  triangle  A C  ( 

Pis  equall  to  the  whole  triangle  C F  G.Where- 
fore the line  AG  is  a  parallel  vnto  the  line  CP 
(by  the  38.propofiti6)-.for  the  triangles  confifl 
vpon  one  and  the  felfe  fame  bafe,  namely  C  P. 

Againe  forafmuch  as  either  of  thefe  angles  AF 
G  Sc  CBG is  the  halfe  ofaright  angle, therfore 
{  by  the  1 7  .propofition  jthe  line  AF  is  a  paral¬ 
lel  to  the  line  C  G.  Wherfore  the  right  line  AF  is  equal  to  the  right  line  C  G( for  the  op- 
pofite  fides  of  a  parafielograme  are  equal! ).  And  forafmuch  as  there  are  two  triangles 
A'BF  and  B  C^.whofe  alternate  angles  are  equali,namely,  the  angle  A  FB  to  the  an¬ 
gle  BGC,  and  the  angle  B  AFto  the  angle  B  C  G’,and  one  fide  of  the  One  is  equall  to 
one  fide  of  the  other,namely,the  fide  which  lieth  betweue  the  equal  angles,that  is,  the 
fide  A  F  to  the  fide  CG,  therefore  (by  the  26.  proposition)  the  fide  A  Bis  equal  to  the 
fide  B  C,andthe  fide  B  F  to  the  fide  B  G.  Wherefore  it  is  proued  that  the  fquares  of  the 
finest  P  andCC  being  equalhtheir  fides  alfo  llialbe  equall:  which  was  required  to  be 
proued. 


he 33.  Theoreme*  ^The 4.7. 'Profofition. 

In  rectangle  triangles  y  the  fquare  whiche  is  made  of  the  fide 
that  fubtendeth  the  right  angle Js  equal  to  the fquares  which 
are  made  of  the fides  containing  the  right  angle. 


^'Vppofe  that  ABC  be  a  reSlangle  triangle,  hauyng  the 
angle  BAC  a  right  angle. The  I  fay  y  the  fquare  Tubieh 
is  made  of  the  line  BC  is  equall  to  the  fquare  s'fthkb  are 
]made  of  y  lines  AB  and  JiC.  Defcribe(by  y  ^6.propoJti 
cion )  Vpotiy  line  BCa  fquare BBCE^and  (byy  fame) 
Vpon  the  lines  'BA  and  AC  deferibe  the fquares  ABEG 
and  AC  f^H.Jnd  by  the  point  A  draH>  (  by  the  pro * 
'pofttion )  to  either  of  thefe  lynes  BID  and  CE  a  parallel 

line 


ofEuclides  Elements s. 


F0/.58. 


line  A LtAnd  (by  the  firEt  petition)  draw  a  right  If  ne  from  the  point  A  to  th  e 
point  B)yand  an  other  from  the  point  C  to  the  pomt  KAnd  fora /much  as  the  an* 
gles  B  A  Cand  B  A  G  are  right  angles ,therf ore  Vntoa  right  line  B  Ay  and  to  a 
point  initgeuen  Ay  aredrawen  two 
right  lines  A  C  and  AGynot  both  on 
one  and  the  fame  fide ,  makyng  the 
two  fide  angles  equall  to  two  right 
angler.  wher  for  e(by  the  14.  propofi * 
tion)the  lines  A  C and  A  G  make  di* 
re  Elly  one  right  line.  And  hy  the  fame 
reafon  the  lines  B  A  and  A  H  make 
alfo  dire  Elly  one  right  line ,  And  fora 
afmtich  as  the  angle  B)BC  is  equall 
to  the  angle  FBA  ( for  either  of  the 
is  a  right  angle)put  the  angle  A  B  C 
common  to  them  both :  wherfore  the 
"whole  angle  D  BA  is  equal!  to  the 
' whole  angle  F  B  CAnd  forafmuch  as  thefe  two  lines  A  B  and  B  B)  are  equal  to 
thefe  two  lines  B  F  and  B  C,the  one  to  the  other ,and the  angle  t>B  A  is  equal 
to  the  angle  FBC:  therfore(by  the  ^.propofitionfihe  bafe  A  B)  is  equall  to  the 
bafe  F  Cyand  the  triangle  ABB)  is  equal!  to  the  triangle  FB  C,  But  (by  the  ji. 


lellynes,  that  isyBB)and  A  Land  (by  the fame)  the / quart  GBit  double  to 
the  triangle  FBC,  for  they  haue  both  one  and  the  felfe  fame  bafe ,  that  is y  B 
F,  and  are  in  the felfe fame  parade  Hynes ,  that  is,  FB  and  G  C.  But  the  dou* 
hies  oft  hinges  equal f  are  (bythejixte  common  f entente)  equall  the  one  to 
the  other.  Wher fore  the parallelograme  B  L  is  equall  to  the  fquare  G  B%  And 
in  like forte  if (by  the firft  peticion )  there  be  drawen  a  right  line  from  the  point 
A  to  the  point  E^and  an  other  from  the  point  B  to  the  point  We  may  prone  y 
the  parallelograme  CL  is  equal  to  the fquare HCyVher fore  the  whole fquare 
BB)  EC  is  equall  to  the  two fquares  G  B  andHC.But  the  fquare  BP  EC  is 
defcribed  Vpon  the  line  B  Cymd  the  fquares  G  B  and  HC  are  defcribedvppon 
the  lines  BA^rAC:  wher  fore  the  fquare  ofthefideBCjs  equal  to  the  fquares 
of  thefidesBAand  A  CyVhereforein  reElangle  triangles,  the  fquare  whiche 
is  made  of  the  fide  that  fubtendeth  the  right  angle  js  equal  to  the  fquares  which 
are  made  of  the  j ides  contayning  the  right  angle:  which  was  required  to  be  de* 
monfirated. 

This  moft  excellent  and  notable  Theoremc  was  firft  inacntedof  the  create 
philofopher  Pithagorasawho  for  the  exceeding  icy  concerned  of  the  indention 
therofjofferedin  facrificcan  Oxeaas  recorde  Hierone,  Proclus,  Lycius,&  Vi- 
trmmis.And  it  hath  bene  commoly  called  of  barbarous  writers  of  th"  latter  time 
Dulcarnon. 

Q.ji.  An 


I’ithagoras  the 
firft  snuenter  of 
this propofttion. 


An  addition  of 
Pektarius. 


An  other  adit  to 
of  Pelstarius. 


Another  additi 
on  of -Pelt  farms. 


Thefir/lTloofy 

An  addition  of  F  elitarius. 

To  reduce  two  vncquall  fquares  to  two  equall  fquares* 

Suppofe  that  the  fquares  of  the  lines  tA  B  and./!  C  be  vnequall.lt  is  required  to  re¬ 
duce  them  totvoequallfquares./oynethetwolinese^Sand^Cat  theirendes  in 
fuch  fort  th  at  they  make  a  right  angle  B  A  C.And  draw  a  line  from  B  to  C.  Then  vppon 
the  two  endes  B  and  C  make  two  angles  eche  of  which  may  be  equal  to  halfe  a  right  an¬ 
gle  (This  is  done  byereftingvpon  the  line  3  Cperpe- 
diculer  lines.from  the pointes B and C :  and fo  (by  the 
p.  propofition )  deuiding  eche  of  the  right  angles  into 
two  equall  partes) :  and  let  the  angles  3  CD  and  CBD 
beeither  of  the  halfe  of  a  right  angle.  And  let  the  lines 
B  D&ndCD  concurre  in  the  point  D.  Then/fav  that 


© 

the  angle  at  the  pointe  D  is  (by  the  3  2.  propofition)  a 
right  angle.  Wherefore  the  fquare  of  the  fide  3  Cise- 

qual  to  the  fquares  of  the  two  fides  D  B  and  D  C  (by  the47.  propofition)  :butitis  alfo 
equall  to  the  fquares  of  the  two  fides  A  Bund  A  C(by  the  felffame  propofition)  wher- 
forer  by  the  common  fentencejthe  fquares  of  the  two  fides  B  D  and  D  Care  equall  to 
the.fquares  of  the  two  fides  ABztxdA  C ;  which  was  required  to  be  done. 

An  otheradditionofPelitarius* 

If  mo  right  angled  triangles  haue  equall  bafes.  the  fquares  of  the  tVeo  fides  of  the  one  are  equall 
to  the fquares  oft  he  ttyo  fides  of  the  other. 

1  his  is  manifeft  by  the  former  conftmftionand  demonftration* 

An  other  addition  of  pelirarius. 

Two  vnequall  lines  beinggeuen,  to  knoty  hoW  much  the  fquare  of  the  one  is  greater  then  the 

fquare  of  the  other , 

Suppofe  that  there  be  two  vnequal  lines  AB  and  B  C.-of which  let  A3  be  the  grea¬ 
test  is  required  to  fearch  out  how  much  the  fquare  of  A  B  excedeth  the  fquare  ot'3 
C.Thatis  I  wil  finde  out  the  fquare,  which  with  the  fquare  of  the  line  B  C  lhalbe  equal 
tothefquare  ofthelinev!  .8.  Put  the  finest 
Aand^Cdircftly,that  they  make  both  one 
righ  t  line.-and  making  the  centre  the  point  B, 
and  the  fpace  BA  deferibe  a  circle  ADE,&txd 
produce  the  line  tAC  to  the  circumference, 
and  let  it  concurre  with  it  in  the  point  f’.And 
vpon  the  lyne  A  E  and  fro  the  point  C  ere& 

(bv  the  1 1. propofition)  aperpendiculerline 
CD, which  produce  till  it  concurre  with  the 
circumference  in  the  point!):  &  drawaline 
from  B  to  Z>.Then  I  fay,that  the  fquare  of  the 
line  CD, is  the  excefle  of  the  fquare  of  the  line 
A  B  aboue  the  fquare  of  the  line  B  C.  For  for- 
afmuch  as  in  the  triangle  8CjD,theangle  at 
the  point  C  is  a  right  angle,  the  fquare  of the 
bafeZ?D  is  equall  to  the  fquares  of  the  two 
fides  B  Cand  C  D(by  this  47.  propofition).Wherefore  alfo  the  fquare  of  the  line  A  B 
is  equall  to  thefelfe  fame  fquares  of  the  lines  rBC  and  CD.  Wherefore  the  fquare  of 
the  line  B  C  is  fo  much  leflfe  then  the  fquare  of  the  line  A  B,  as  is  the  fquare  of  the  line 
C D;  which  was  required  to  fearch  out,  Aq 


the  two  fquares  of  the  fides  B  D  and  C  D,  are  equall  to 
the  two  fquares  of  the  fides  A  B  and  A  C.For  (by  the  6 
propofition)the  two  fides  and D  Carceauall  and 


Fol^p. 


ofEuclides  Elements* . 

An  other  additionofPelitadus. 

T he  diameter  of  a  fij  Hare  bcinggeuenjtogcM  the fquare  thereof  An  other  aditio 

of  Pehtarius, 

This  is  eafie  to  be  done.  For  ifvpon  the  two  endes  of the  line  be  drawen  two 
halfe  right angles,and  fo  be  made  perfed  the  triangle  then  flialbe  defcnbed  half 
of  the  fquarejthc  other  halfe  whereof  alfo  is  after  the  fame  manner  eafie  to  be  de? 
fcribed. 

Hereby  it  is  manifejfthat  the  fquare  of  the  diameter  is  double  to  that Square  Vphofe  diameter  it  is.  ^  Corroltarj 

The  i\.Theoreme.  The  4.8.  Tropofition \ 

If  the  fquare  which  is  made  of one  of  the fides  of  a  triangle? 
he  equal!  to  the  fquare s  which  are  made  of  the  two  other  fides 
of  the  fame  triangle:  the  angle  comprehended  vnder  thofe 
two  other  fides  is  a  right  angle . 

“  Vppofe  that  ABC  be  a  triangle, and  let  the  fquare  "tobich  is  made  of one 

!  of  the  fides  there  jiamely  yof the fide  B  Cfe  squall  to  the  fquares  which 
are  made  of  the  fides  BA  and  A  C.Then  I fay  that  t  he  angle  B  AC  is  a 
right  angle.  3(yyfe’vp(by  the  u.propofitio) from  the  point  Avnto  the  right  line 
A  C  a  perpendicular  line  A  V,  And  (by  the  thirde  propofition)  Vnto  the  line  A 
B  put  an  equall  line  A  3),  And  by  the  firfl  petition  draSo  a  right  line  from  the 
point  V  to  the  point  C.  And for af much  as 
the  line  3)  A  is  equall  to  the  line  A  B«  the 
fquare  Tbhich  is  made  of  the  line  3)  A  is  e* 
quail  to  the  fquare  Srhiche  is  made  of  the 
line  A  B  But  the fquare  of  the  line  A  C, 
common  to  them  both .  Wherefore  the 
fquares  of  the  lines  3)  A  and  AC  are  equal 
to  the  Jquares  of  the  lines  BA  and  AC, , 

But  (by  the  propofition  going  before)  the 
fquare  of  the  line  3)C  is  equal  toy  fquares 
of  the  Imes.AV  and  A  C.  (For  the  angle 
VAC  is  a  right  angle)  andthe  fquare  of 

B  C  is  (by  fupp  option )  equall  to  the  fquares  of  A  Band  AC,  Wherefore  the 
fquare  of  3)  C  is  equal!  to  the  fquare  ofB  C  therefore  the  fide  V  Cis  equall  to 
the  fide  BC,  And  (ora/much  as  A  B  is  equall  to  A  V  and  AC  is  common  to  them 
loth,  therefore  the fe  two  fides  V  A  and  AC  are  equall  to  thefe  t^o  fides  BA 
and  A  C,  the  one  to  the  other,  and  the  hafe  V  C  is  equall  to  the  hafe  B  C\St>her - 
fore  (by  the  propofition)  the  angle  1)  ACis  equall  to  the  angle  B  A  C  But  the 
angle  VAC  is  a  right  angle  therefore  atfo  the  angle  B  AC  is  a  right  angle.  If 

Q.dtj.  there * 


This  propofition 
it  the  Conner  (e 
of  the  former. 

*in  other  De- 
monflration  af¬ 
ter  Pelt  tart  us. 


cThefirHcBooke 

therefore  the  fquare  Tbhich  is  made  of  me  of  the  fides  of  a  triangle^  he  equal 1  to 
the  fquaresfbhicb  are  made  of the  ftoo  other fides  of  the  fame  triangle ,  the  an * 
gle  comprehendedvnder  thoJetTro  other Jide sis  a  right  angle  :  "tohich  was  re* 
qniredtohe  proued.  - 

This  propofition  is  the  conuerfeofthe former,*  d  is  of  Pelitarius  demon* 
ftrated  by  an  argument  leading  to  an  impoffibilitie  after  this  maner* 

S  uppGfe  that  ABC  be  a  triangle :  &  let  the  fquare  of  the  fide  AC, be  equal  to  the  fquare  s 
of  the  two  fides  A  B  and  B  C.Then/ fay  that  the  angle  at  the  point  ^which  is  oppofite 
to  the  fide  AC  yis  a  right  angle. For  if  the  angle  at  the  point  c 

B  be  not  a  right  angle,  then  fhal  it  be  eyther  greater  or  lefle 
the  a  right  angle.  Firft  let  it  be  is  greater.  And  let  the  angle 
DBC  be  a  right  angle,  by  erecfing  from  the  point  B  a  per¬ 
pendicular  line  vnto  the  line  BC( by  the  n, propofition) 
which  let  be  B  D:  and  put  the  line  #£>  equall  to  the  lyne 
A  B  (by  the  thirde  propofition  And  drawe  a  line  from  Q 
to  D.  Now  (by  the  former  propofition)  thefquare  of  the 
fide  CDfhalbe  equall  tothefquares  of  the  two  fides  BD 
a nd#  C :  wherefore  alfo  to  the  fquares  of  the  two  fides  B 
yjfand  BC.  Wherefore  the  bafe  CD  fiialbe  equall  to  the 
bafe  C  A,  when  as  their  fquares  are  equall :  which  is  con¬ 
trary  to  the  24.propofition.Forforafmuchasthe  angled 
BCis  greater  then  the  angle  DBC,  and  the  two  fides  A  B 
and  BC  are  equall  to  the  two  fides  D  B  and  B  C,  the  one  to  the  other,  the  bafe  C  A  (hall 
be  greater  then  the  bafe  CD,  It  is  alfo  contrary  to  the  7.propofition,  for  from  the  two 
endes  ofone  &  the  fame  line,  namely,  fro  the  points  B  &  Cihould  be  drawn  on 
one  and  the  fame  fide  two  lines  B  D  and  D  C  ending  at  the  pointe  D,  e- 
quall  to  two  other  lines  BA  and  A  Cdrawen  From  the  fame  endes 
and  endin  g  at  an  other  point,  namely,at  A, which  is  impof- 
fible.By  the  fame  reafon  alfo  may  we  proue  that  the 
whole  angle  at  the  pointe  B  is  not  lefle  then  a 
right  angle.  Wherfore  it  is  a  right  angle: 
which  was  required  to  be  proued. 

(’••) 

The  endc  of  the  firft  hoo^e  ofSucl'tdes  Slementes , 


€j  The  fecond  booke  of  Eu- 


60 * 


elides  Elementes* 


N  this  fecond  booke  fidclide  ilie^etK ,  t-ha^  is  a 
Gnom6,anda  right  angled  parallelogramme.  Alfo 
in  this  bobke are  let  forth  the  powers  cilines,deui- 
ded  euenly  and  vneuenly , arid  of  lines  added  one  to 
an  other.  Thepower  ofaline ,  is  the  fquare  of  the 
fame  line:  tffttis,  afqitat^  euery  fide  of  which  is  e- 
quail  to  tire  line*  So  that  here  are  fct  forth  the  quali¬ 
ties  and  proprieties  of  the  fquarcs  and  right  lined  fi- 
guresjwntch are  made oilines  &  oftheir  parts.  The 
Arithmetician  alfo  out  ol  this  booke  gathered!  ma? 
ny  compendious  rules  of  reckoning, and  many  rules 
alfo  of  Algebra, with  the  equatios  therein  vied.  The 
groundes  alfo  ofthoie  rules  are  for  the  m  oft  part  by  this  fecond  booke  dernon- 
ffrated*  This  booke  moreouer  contayncth  two  wondbrfull  propoficions*  one  of 
an  qbtufe  angled  triangle,  andthe  other  ofan  acuterwhich  with  the  ayde  of  the 
47*propofition  ofthefirft booke ofEuclide,  which  isofa  re&angle  triangle,ol 
how  great  force  and  profite  they  are  in  matters  ofaftronomy,they  knowe  which 
haue  trauayled  in  that  arte*  V  Vherefore  if  this  booke  had  none  other  profite  be 
fide*  onely  forthcfez*prapofitions  fake  it  were  diligently  to  be  cmbracedand 
ihidied. 


The  argument 
of  the Jecond 
books • 

fVhat  it  the 
poTver  of  a 
tine . 


Manycompe * 
dious  rules  of 
reckoning  ga- 
theredout  of 
this  bookstand 
aljo  many 
rules  of  Alge¬ 
bra* 

Ttpo  Wonder* 
full  proporti¬ 
ons  in  this 


ejimtions . 


i.  Suery  reBangled parallelogramme,  is [ayde  to  be  contayned 
vnder  two  right  tines  comprehending  a  right  angle* 

A  parallelogramme  is  afigure  effower  fi  Jes,whofe  two  oppofite  or  contra?  hatapa * 

ry  fides  are  equall  the  one  to  the  other.  There  are  of  paralldogrammcs  fower  ™Uelogrammf 
kyndes,afquareJafigureofonefidelonger,aRombusordiamond,andaRom-  ,  . 
boides  or  diamond  like  figure,as  before  was  fayde  in  the  ^.definition  of  thefirif  otparaMod* 
booke.  Ofthefe fower  fortes,  the  fquarc  andthefigure  of  one  fide  longer  are  grammes* 
onely  right  angled  Parallclogrammes;  for  that  all  their  angles  are  right  angles. 

And  either  of  them  is  contayned  (according  to  this  definition )  vnder  two  right 
ly  ties  which  concurre  together ^nd  caufe  the  right  angle,and  concaine  the  iame. 

Of  which  two  lines  the  one  is  the  length  of  the  figure,  &  the  other  the  breadth. 

The  parallelogramme  is  imaginedto  be  made  by  thedraught  or  motion  ofone 
efthe  lines  into  the  length  ofthe  other,As  if  two  numbers  ihoulde  be  multipli¬ 
ed  the  one  into  the  other*  AS  the  figure  AB  C  D  is  a  parallelograme,  and  is 
fayde  to  be  contayned  vnderthetwo  right  lines  A  B  and  A  C,which  contayne 
the  right  angle  B  A  C,or  vnder  the  two  right  lines  AC  and  j  ^ 

C  D,  for  they  likewife  contayne  the  right  angle  A  C  D:  of  ~ ' 1 — — “ — 
which  Julfnes  the  one,namely  ,A  B  is  the  length ,  and  theo- 
ther,namely,AG  is  the  breadth*  And  ifwe  imagine  the  line  c  ~  v 
AC  tobe  drawen  or  moueddir e&ly  according  to  the  legth 

Q»uii,  q£ 


* 


Stmddt- 

fimtioHt 


r  ?  *  The/econd^Boo^e 

of  the  line  A  B,or  contrary  wife  the  line  A  B  to  be  moiled  dire&ly  according  to 
the  length  of  the  line  AC,  you  fhall  produce  the  whole  re&anglc  parallelo- 
gramme  AB  CDwhichisfaydctobccontaynedofthem:  eucn  as  one  number 
multiplied  by  another  produccth  a  plaine  and  rightc  angled  fuperficiall  num¬ 
ber, as  yc  fee  in  the  figure  here  fet,  -where  the  number  of  fixe  *  % 

or  fixe  unities,  is  multiplied  by  the  nupiber  of  fiuc  or  by 
flue  vnities:  ofwhich  multiplication  are  produceijo^which 
number  being  fet  downe  and deferibed  by  his  vn ides  repre- 
fenteth  a  play  ncanda  right  angled  nurr^ter*  Wherefore  c- 
ucn  as  cquall  numbers  multiplcd  by  cqua^l  numbers  produce 
numbers  cquall  the  one.  to  the  other: fo reaangle  parallelo- 
grames  which  are  comprehended  vnder  equal  Lines  are  equal 
the  one  to  the  other. 


3o 


$ 


.  . 

1  ■  S.‘/tL  ■ 

.  . 

*  ; ; 

-  *-  "  ♦  '  .  : 

*  * 


2. 


jl  ’Jr,  mu 

n-J 

■ft 


In  euery parattelogramme ,  one  of thofi  parallelogrammes , 
which  foeueritbe,  which  are  about  the  diameter ,  together* 

•  11  s-  ,  •  ft  t  V-V  ‘  - 

lementesyi 


nomon. 


■1 


Thofe  perticulcr  parallelogramcs  are  faydeto  be  about  the  diameter  of  the 
parallelograme, which  hauc  the  fame  diameter  which  the  whole  parallelograme 
hath^And  fupplementesareluchjwhich  arc  without  the  diameter  ofthe*  whole 
parallelograme.  a  s  of  the  parallelograme  ABCD  the  partial  or  perticulef  paral  - 
lelogrames  GKCF  and  E  B  K  H  are  parallelogramcs  about  the  diameter,  for 
that  ech  ofehem  hath  for  his  diameter  a  part  o.f  the  diameter  ofthe  whole  paraL* 


lelogramme.  As  C  K  and  K  B  the  pcrticuler  diameters,  are  partes  of  the  line 
C  B,which  is  the  diameter  ofthe  whole  parallclogramme.And  the  two  paralle- 
logrammes  A  E  G  K  and  KHF  D,are  fupplcmentes,becaufe  they  are  wy  thout 
the  diameter  ofth  c  whole  parallelogramme,namely ,C  B.Now  any  one  ofthofe 
partiall  parallelogram mes  about  the  diameter  together  with  the  two  fupple- 
mentes  make  a  gnomon.  As  the  parallelograme  EB  K  H,  with  the  two  fupple- 
mentes  A  EG  K  andK  HF  Dmake  the  gnomon  FGEfL  Ltkewifetheparal* 
lelogramme  G  K  C  F  with  the  fame  two  fupplemcntcs  make  the  gnomon  E  H 
F  G.Andthtsdiffinitionofagnomonextendethitfclfc,  and  is  general!  to  all 
ky  tides  of  parallelogrammes, whether  they  be  fquarcs  or  figures  of  one  fide  lon¬ 
ger  or  Rhombus  or  Romboides.  To  be  fliorte,ifyou  take  away  from  the  whole 
parallelogramme  one  ofthe  partiall  parallelogrammes  which  are  about  the  di¬ 
ameter 


Fol.6  f. 


ofEucUdes  Elementes. 


amctet  whether  ye  wilijthe  reft  of  the  figure  isa  gnomon. 


Campa  eafterthelau  propofitionof thefirftbookeaddeth  this  propofitio. 
Tvpo  Squares  bemggeuen ,  to  adioyne  to  one  of  them  a  Gnomon  equali  to  the  other fquare  .-which,  for 

that  as  then  it  was  not  taught  what  a  Gnomon  is,1  there  omitted,  thinking  that 
it  might  more  aptly  beplacedherc.The  doing  and  demonftration  whereat,  is 
thus . 


proportion 
aided  by  (am* 
pane  after  the 
tajl  proporti¬ 
on  of the firfk 
hooks* 


Suppofe  that  there  be  two  fquares  A  B  and  C  D:  vnto  one  of  which,  namely ,  vnto 
A  £, it  is  required  to  addc  a  G  nomon  equali  to  the  other  fquare,  namely,  to  C  lD .  Pro¬ 
duce  the  fide  B  F  of  the  fquare  AB  di- 
reftly  to  the  point  £.and  put  the  line  F 
£  equali  to  the  fide  of  the  fquare  CD. 

And  draw  a  line  from  E  to  A.  Now  then 
forafmuch  as  E  F  Ais  a  rediangle  trian- 
gle,therefore(by  the  47.  of  the  firft)  the 
fquare  of  the  line  EA  is  equali  to  the 
fquares  of  the  lines  EF  &  FA.  But  the 
fquare  of  the  line££  is  equali  to  the 
fquare  C2>,&  the  fquare  of  the  fide  FA 
is  the  fquare  A  ^.Wherefore  the  fquare 
of  the  line  AE  is  equal!  to  the  two  fquares  CD  and  A  £.But  the  fides  E  F  and  F  A  arc 
(by  the  ai.  of  the  firftj  longer  then  the  fide  <sA  £,andthe  fide  F  A  is  equali  to  the  fide 
£  £.  W herfore  the  fides  £  F  and  FB  are  longer  the  the  fide  A  £♦  Wherefore  the  whole 
line  BE  is  longer  then  the  line  A  £,From  the  line  £  £  cut  of  a  line  equali  to  the  line  A 
£,which  let  be  B  C.And  (by  the  q-tf.propofition )  vpon  the  line  B  C  deferibe  a  fquare, 
which  let  be  BCGHtwhich  lhalbe  equal  to  the  fquare  of  the  line  A  £,but  the  fquare  of 
the  line  A  E is  equal  to  the  two  fquares  A  B  andD  C.Wherefore  the  fquare  B  CGH  is 
equal  to  the  fame  fquares.  Wherfo  re  forafmuch  as  the  fquare  BCG  His  compofed  of 
the  fquare  o 4  B  and  of  the  gnomon  £  G  A  H ,  thefayde  gnomon  flialbe  equali  vnto 
the  fquare  C‘D:which  was  required  to  be  done. 


An  other  more  redy  way  after  Pelitari  us* 

Suppofe  that  there  be  two  fquares,whofe  fides  let  be  iAB 
and£  C.It  is  required  vnto  the  fquare  of  the  line  <*^££,to  adde 
a  gnomon  equali  to  the  fquare  of  the  line's  C.Setthc  lines 
B  and  B  Cin  fuch  fort  that  they  make  a  right  angle  ABC,  And 
draw  a  line  fro to  C.And  vpo  the  line  AB  deferibe  a  fquare 
which  let  be  A  B  ‘Z>  £,And  produce  the  line  B  A  to  the  point 
£,and  put  the  line  BF  equali  to  the  line  AC,  And  vpon  the 
line  B  F  deferibe  a  fquare  which  let  be  B  F  G  H :  which  flialbe 
equal  to  the  fquare  of  the  line  A  C,whe  as  the  lines  B  F  and  A 
Care  equal  land  therefore  it  is  equal  to  the  fquares  of  the  two 
hnes^Aand  £C  N°w  forafmuch  as  the  fquare  BFG  His  made  complete  bv 
the  fquare  A  £  D  £  and  by  the  gnomon  £  £  GD,the  gnomon  F  E  CD  flialbe  * 
equal  to  the  fquare  of  the  line  £  Cjwhich  was  required  to  be  done. 


Conjlruttm, 


Vemottjlratio 


The feconcffiookg 

§&Thei  fit heoreme .  The  iSPropofition* 

ff there  be  typo  right  lines ,  and  if  the  one  of  them  be  deuided 
into  partes  hovoe  many foeuer  :  the  reBangle figure  compre - 
hended vnder  the  two  right  lines  js  e quail  to  the  reBangle y£- 
gures  yphiche  are  comprehended vnder  the  line  vndeuided% 
and  vnder  euery  one  of  the partes  of  the  other  line . 


HH  Vppofe  that  there  be  typo  right  lynes  oyf 
and  3  C  and  let  one  of  them y  namely  y  3  Che  deui* 
dedatalladuenturesin  the  pointesDand  E.Then 
I/ay  that  the  reBangle  figure  comprehended  Vn» 
jder  the  lines  A3and  3  Cyis  e  quail  Vnto  the  re  Ban * 
if*  figure  comprehended  Vnder  the  lines  Jyand  3 
'~D,oVnto  the  reBa?igle  figure  which  is  coprehen » 
ded  Vnder  the  lines  A  and  3)  Ey  and  alfo  Vnto  the 

_  ^reBangle  figure  which  is  comprehended  Vnder  the 

lines  A  and  E  C,  For  from  the  point  e  3rayje  Vp  (by  the  u.of the  firfi)  vnto  the 
right  line  3C  a  perptndiculer  line  3  EyO  Vnto  the 
line  A  (by  the  third  of  the fir  It)  put  the  line  3Ge* 
quail ,  and  by  the  point  G  (by  the  $  1 ,  of  the  firfi ) 
draw  a  parallel  line  Vnto  the  right  line  3  C  and  let 
the  fame  be  G  Myand(by  the  felfe fame)  by)  points 
DyE^andCy  draw  Vnto  the  line  3  G  the/e  parallel 
lines  ID  E  L  and  C  H.  TSloW  then  the  parallelo- 

grame3  His  equallto  thefe  parallelogrammes  3 
KJD  L}and  E  H.3ut  the parallelograme  3  His 
equall  Vnto  that  which  is  contayned  Vnder  thelinesAand3C.  (For  it  is  com* 
preheded  Vnder  the  lines  G3  O  3Cyand  the  line  G  3  is  equall  Vnto  the  line  A) 
And  the  parallelograme  3  Kjs  equall  to  that  which  is  contayned  Vnder  the  lines 
A  and  3  D:  (for  it  is  comprehended  Vnder  the  line  G  3  and  3  D}and  3Gise» 
quail  Vnto  A)  And  the  parallelograme  D  L  is  equall  to  that  winch  is  contayned 
Vnder  the  lines  A  and  D  E(for  the  line  D  that  is/3  Gis  equal  Vnto  A) And 

moreouer  likewife  the  parallelograme  E  His  equall  to  that  which  is  contained 
Vnder  the  lines  A  o  EC.  VVberfore  thd  t  whtch  is  compreheded  Vnder  ji  lines  A 
O  3C is  equall  to  that  which  is  comprehended  Vnder  the  lines  A  O' 3  D-,0  Vn* 
toy  which  is  compreheded  Vnder  the  lines  A  and  D  Ey  and  moreouer  Vnto  that 
which  is  comprehended  Vnder  the  lines  A  and  E  CJf  therfore  there  be  two  right 
lines  }and if \ the  one  of them  be  deuided  into  partes  how  many  foeuer  yt  he  reBan* 


Fol.dl. 


ofEuclides  Elementes. 

gle  figure  comprehended  Vnder  the  ttoo  right  lines  js  squall  to  the  re&angle  fi* 
gures  ‘tohich  are  comprehended  Vnder  the  line  Vndeuided  and  Vnder  euery  one 
of  the  partes  of  the  other  linei'tobicb  ‘tods  required  to  be  demonstrated* 

Becaufe  that  all  the  Proportions  of  this  fecond  booke  for  the  moft  part  are 
true  both  in  lines  and  in  numbers,  and  may  bedeclaredby  both:  therefore  haue 
I  haue  added  to  euery  Propofition  conuenient  numbers  for  themanifeftation  of 
the  fame.  Andto  the  end  the  ftudiousand  diligent  reader4  may  themore  fully 
perceaue  and  vnderftand  the  agrementof  this  art  ofc  Geometry  -with  the  fcience 
of  Arithmetiquejandhow  nere  Sc  deare  fillers  they  are  together,fo  that  the  one 
cannot  without  great  blemifh be  without  theother,  1  haue  here  alfo  ioyneda 
little  booke  of  Arithmetique  written  bv  on c'Barlaam,  a  Greekc  authour  a  man 
of greate  knowledge*  In  whiche  booke  arc  by  the  authour  demonftrated 
many  of  the  felfe  fame  proprieties  andpafihons  in  number,  which  Euclideia 
this  his  fecondboke  hath  demonftrated  in  magnitude, namely  .the  firft  ten  pro- 
pofitidns  as  they  follow  in  order.  Which  is  vndoubtedly  great  pleafure  to  co- 
fider,alfo  great  increafe  SC  furniture  ofkno  wledge.  Whole  P  ropofitiSs  are  fee 
orderly  after  the  propofitiSs  of Euclids,  euery  one  of^/^iwcorrefpodent  to  the 
fame  o£Euclide.An&  doubtles  it  is  wonderful  to  fee  howthefe  two  cotrary  kynds 
of  quantity ,  quantity  diferete  or  number, and  quantity  continual  or  magnitude 
(whicharethe  fubie&es  or  matters  of  Arithmitique  and  Geometry  )  fhoulde 
haue  in  them  oneand  the  fame  proprieties  common  to  them  both  in  very  ma¬ 
ny  pomts.and  affc<ftions,although  not  in  all. For  a  line  may  in  fuch  fort  be  dc- 
uided,  that  what  proportion  the  whol  e  hath  to  the  greater  parte  the  fame  {hall 
the  greater  part  haue  to  the  leCIc^  But  that  can  not  be  in  number.  For  a  number 
cannot  fo  be deuidedjthat.the  whole  number  to  the  greater  part  thereof,  ftiall 
haue  that  proportion  which  the  greater  part  hath  to  the  leffe,  as  lordanevery 
playnely  protieth  in  his  booke  of  Arithmetike,  which  thynge  Campane 
alfo  fas  we  ftialiafterwardinthe9*  booke  after  the  15*  propofition  fee)  proueth* 
Andas  touching  thefe  tenne  firfte  propofinonsof  the  feconde  booke  of  Eu- 
clide,demonftratedby  Barlaam  in  numbers,they  are  alfo  dembftrated  of  Cam- 
pane  after  the  i^propofition  ofthe9- booke,  whofe  demonftrations  I  mynde 
by  Godshelpe  to  fetforth  when  I  fiiai  come  to  the  place.  They  are  alfo  demo- 
llrated  of  lordane  that  excellet  learned  authour  in  the  firft  booke  of  his  Arith¬ 
metike. Inthc  meane  ty  me  I  thougheit  not  amilfe  here  to  fet  forth  the  demon- 
ftrations  of  Barlaam, for  that  they  geue  great  light  to  the  feconde  booke  ofEu- 
clide,befides  the  ineftimable  pleafureywhich  they  bring  to  the  ftudious  confidc- 
rer,Andnowto  declare  the  firft  Propofition  by  numbers.  I  haue  put  this  exam¬ 
ple  following. 

Take  two  numbers  the  one  vndeuided as  74, the  other  deuided  into  what  partes 
37*  deuided  into  20. 10.?.  and  2:which  altogether  make 
the  whole  57.  Then  ifyou  multiply  the  number  vndeuided,  namely,  74,  into  all  the 
partes  ofthe  number  deuided  as  into  20. 10.  ?.and  2.  you  (hall  produce  1480.  740* 

3  70  .  i^S.which  added  together  make  273  8  :which  felf  numberis  alfo  produced  if  you 
multiplye  the  two  numbers  firft  geuen  the  one  into  the  other.  As  you  fee  in  the  exam¬ 
ple  on  the  other  fide  fet. 


Barlaam, 


Barham, 


The fecondBoofy 


74 

Multiplication  of  the  whole 

1480 

nuber  vndeuided  into  the 

740 

partes  of  the  whole  num- 

370 

ber  deuided. 

148 

•s 

2738 

- 

Multiplication  of  the  one 

74 

whole  number  into  the  0- 

3  7 

ther.  y/e 

L  '  ?  -y 

518 

X  * 

t "  7  "l—  ” 

-  2  7  3  8 

5 


2 

the  number  produced  of  the  one-s 
whole  number  into  the  partes  of 
the  other  whole  number 

^equall  to 


^the  number  produced  of  the 
fame  whole  into  the  other  whole  - 


So  by  the  aide  of  this  Propofition  is  gotten  a  compendious  way  of  multiplication  by 
breaking  ofoneof  the  numbers  into  his  partes:  which  oftentimes  ferueth  to  great  vie 
in  working, chiefly  in  the  ruleofproportions.Thedemonftration  ofwhich  propofition 
followeth  in  Barlaam.But  firft  are  put  of  the  author  thefe  principlesfoIlowin<*, 

<7  (principles. 

1 .  <zA  number  is fayd  to  multiply  an  other  number:  When  the  number  multiplied  is fo  oftentymes 
added  to  it  felfe, as  there  be  vmties  in  the  number, Which  multiplied:  Vo  her  by  is  produced  a  certame 
number  Which  the  number  multiplied  meafureth  by  the  unities  Which  are  in  the  number  Which  mul¬ 
tiplied  . 

2 .  And  the  number  produced  of  that  a  multiplication  is  called  apUine  or fuperfciall  number. 

3  .  *s4 fquare  number  is  that  which  is  produced  of  the  mnltiplicatian  of  any  number  into  it  felfe. 

4.  Entry  lejfe  number  compared  to  a  greater  is  fayd  to  be  a  part  of  the  greater, Whet  her  the  lejfe  mea~ 
fare  thegr  eater, or  meafure  it  not. 

5 .  TfumberSiWhome  one  and  the felfe fame  number  meafureth  equally,  that  is, by  one  and  the felfe 
fame  num  beY  are  e  quail  the  one  to  the  othet , 

6.  Numbers  that  are  equemultiplices  to  one  and  the  felfe  fame  number, that  is, Which  contayne  one 
and  the  fame  number  equally  and  alike, are  equall  the  one  to  the  other. 

TbefirH  Proportion, 

T  tya  numbers  b  eynggeucnjfth  e  one  of  them  be  deuided  into 
any  numbers  how  many  foeuer:  the  playne  orfuperficiall number 
Which  is  produced  of  the  multiplication  of  the  tWo  numbers  firft 
geiien  the  one  into  the  other, (hall  be  equall  to  the  ftpcrfciall  tiu- 
bers  Which  are  produced  of  the  multiplication of  the  number  not 
deuided  into  eusry  part  of  the  number  deuided. 

Suppofe  that  there  be  two  numbers  A B  and  C.  And 
deuide  the  number  A B  into  certayne  other  numbers 
how  many  foeuer.as  into  A  D,D  £,and£  B ,  Then  I  fay 
that  the  fuperficiall  number  which  is  produced  of  the 
multiplication  of  the  number  Cinto  the  number  zA  B 
is  equall  to  the  fuperficiall  numbers  which  are  produ¬ 
ced  of  the  multiplication  of  the  number  C  into  the  nu- 
ber  A  'Zhand  of  C into  D  £,and  of  C  into  £  B.  For  let  F 
be  the  fuperficiall  number  produced  of  the  multiplica¬ 
tion  of  the  number  C  into  the  number  A  J3,and  let  GH 
be  the  fuperficiall  number  produced  of  the  multipli¬ 
cation  of  Cinto  Aid  .-And  let  H I  be  produced  of  the 
multiplication  of  Cinto  D  £:  and  finally  of  the  multi¬ 
plication  of  Cinto  FB  let  there  be  produced  the  num-  c  A  F  G 

ber 


ofEuclides  Ekmentes .  Fol.  6^ 

ber/ ANowforafrniichas.^FmuItiplying  the  numberC  produced  the  number  F; 
therefore  the  number  C  meafureth  the  number/7  by  the  vnities  which  are  in  the  nnm- 
ber  A  B.  And  by  the  fame  reafon  may  be  proued  that  the  number  C  doth  alfo  meafure 
the  number  (7 //,by  the  vnities  which  are  in  the  number?^  A  and  that*  it  doth  mea¬ 
fure  the  number  HI  by  the  vnities  which  are  in  the  nuber  D  F  and  finally  that  it  mea¬ 
fureth  the  number  IK  by  the  vnities  which  are  in  the  number  E  B  .Wherefore  the  nu¬ 
ber  C  meafureth  the  whole  number  G  K  by  the  vnities  which  are  in  the  number  AB. 
But  it  before  meafured  the  number  F  by  the  vnities  which  are  in  the  number  ABywher 
fore  either  of  thefe  numbers  F  and  G’F'is  equcmultiplexto  the  number  C .  But  num¬ 
bers  which  areequemultiplices  to  one  &  thqfelfe  fame  numbers  areequall  the  one  to 
the  other  (by  the  tf.definitionJ.Wherfore  the  number  Fis  equadto  the  number  G  K. 
But  the  number  Fis  the  fuperficiall  number  produced  of  the  multiplication  of  the  nu¬ 
ber  C  into  the  number  A  B :  and  the  number  Cj  K  is  ccmpofed  of  the  fuperficiall  num¬ 
bers  produced  ofthe  multiplication  of  the  nuber  Cnot  deuided  into  euery  one  of  the 
numbers  A  D,D  E,andEB.l£ therefore  there  be  two  numbers  geuen  and  the  one  of 
them  be  deuided  &c.  Which  was  required  to  be  proued. 


The  iJTbeoreme.  The  ?. .  Eropofition , 

If  a  right  line  be  deuided  by  chaunce >  the  nil  angles  figures 
which  are  comprehended  vnder  the  whole  and  euery  one  of 
the  partes ,  areequall  to  the  fquare  whiche  is  made  of  the 
whole .  - 

V ppofe  t  hat  the  right  line  AB  he  hy  chaunfe  de- 
nided  in  the  point  C .  Then  1 fay  that  the  reHan * 
file  figure  comprehended  Vnder  Aft  and  B  C  to* 
gether  "frith  the  reH  angle  comprehended  "Vnder 
A  B  and  AC  is  equal!  Vnto  the fquare  made  of  A  B,T)e* 
f cribs  (by  the  4.6,0 f  the firH)  Vpon  A  B  a  fquare  A  T)  E  B: 
and  (by  the  $  1 0}  the  fir  ft)  by  the  point  Cdrafr  a  line  paral * 
lei  Vnto  either  of  tbe/e  lines  A  T>  and  B  E,and  let  the  fame  be  CF.  Tfow  is  the  ^emwJlratiS 
parallelogramme  AE  equallto  the  paralltlbgrammes  A  Fand  C  Ey  by  the  firH 
of  this  books.  But  A  E  is  the  Jquaremadeof  A  B.  And  JF  is  the  re  cl  angle 
parallelogramme  comprehended  "vnder  the  lines  B  A  and  A  C:  for  it  is  compre¬ 
hended  Vnder  the  lines  V  A  and  A  C:  but  the  line  A  V  is  equallvnto  the  line  A 
B,Andlih"frife  the parr allelogramme  C Eis  e quail  to  that  "frhich  iscontayncd 
"Vnder  the  lynes  ABand  B  (for  the  line  BE  is  equal  Vnto  the  line  AB.Wher - 
fore  that  "frhich  is  contayned  "vnder  B  A  and  AC  together  "frith  that  "frhich  it 
contayned  "Vnder  the  lines  AB  and  B  C fis  e  quail  to  the  fquare  made  of  the  line 
A  B  Jf therefore  a  right  line  be  deuided by  chaunce ythe  re  Ha  file  figures  "frhich 
are  comprehended  Vnder  the  "frhole ,  and  euery  one  of  thepartes3are  t  quail  to 
the  fquare  "frhich  is  made  of the  whole: ", frhich  "fras  required  to  be  demonstrated. 

An  other  demonftration  of  Campane. 

...  R  iii.  Sup- 


A  C 

B 

Id  f 

TL 

Conjlruttion, 


’ThefecondBooke 

Suppofe  that  tKe  line  AB  be  deuided  into 
the  lines  AC}  C Z>,and  Di?  .Then  I  fay  that  the 
fquare  of  the  whole  line  A  B, which  let  beAE 
BF,is  equal  to  the  redangle  figures  which  are 
contayned  vnder  the  whole  andeuery  one  of 
the  partes  :  fo r  take  the  line  ICwhich  let  be  e- 
qual  to  the  line  Atf.Nowthen  by  thefirftpro- 
pofition  the  redangle  figure  contained  vnder 
the  lines  A  B  and  /<T,is  equal! to  the  redangle 
figures  contayned  vnder  the  line  K  and  althe 
partes  of  the  line  AB.  But  that  which  is  con¬ 
tayned  vnder  the  lines  K and  A B  is  equall  to 
the  fquare  of  the  line  A  B ,  and  the  redangle 
figures  contay  ned  vnder  the  line  K  and  al  the  ^  G  v  5 

partes  o£AB,  are  equall  to  the  redangle  fi¬ 
gures  contayned  vnder  the  line  A  B  and  all  the  partes  ofthe  line  AB:  for  the  lines 
B  and  K are  equall:  wherefore  that  is  manifefl:  whichwas  required  to  be  proued. 

Anexampleofthis  Propofitionin  numbers. 


Take  a  number, as  1 1  .and  deuide  it  into  two  partesmamely,  7. and  4.’  and  multiply 
x  x  .into  7 ,andthen  into  4, and  there  lhalbe  produced  77. and  44.‘both  which  numbers 
added  together  make  r  2 1  .which  is  equall  to  the  fquare  number  produced  of  the  mul¬ 
tiplication  of  the  number  1 1  .into  himfelfe,as  you  fee  in  the  example, 

•*',  .  .  .  .  ^  *  .  \  ■  . 
ft4'  .  •  t  H  -  ?  *% 


^Multiplication  of  the  whole- 
’  intohispartes. 


A 


|  Multiplication  ofthewhole 
Mnto  himfelfe. 


11 


77 

44 


1 21 


11 
1 1 


11 
1 1 


121 


the  number  produced  ofthe-% 
whole  into  his  partes* 


-equal  to 


'-the  number  produced  of  the. 
whole  into  himfelfe. 


CqT- 


Barlaam. 


The  demo  nftration  whereof  folio  weth  in  Barlaam, 

ThefecondTropofition. 

If  a  number  geuen  be  decided  into  two  other  numbers :  the fuperficiall  numbers ,  Which  arepro* 
disced  ofthe  multiplication  ofthe  Whole  into  either  part, added  together ,are  equall  to  the  fquare  num¬ 
ber  ofthe  whole  number  geuen. 

Suppofe  that  the  number  geuen  be  A  B  : and  let  it  be  deuided  into  two  other  num¬ 
bers  A  C  and  CB.  Then  I  fay  that  the  two  fuperficiall  numbers ,  which  are  produced 
ofthe  multiplication  of  A  B  into  A  C,  and  of  A  B  into  B  C ,  thofe  two  fuperficiall  num¬ 
bers  (I  fay)  beyng  added  together,  lhalbe  equall  to  the  fquare  number  produced  of 
the  multiplicand  of  the  number  AB  into  it  felfe.For  let  the  number  <eA  B  multiplying 
it  felfe  produce  the  number  Df  Let  the  number  A  C  alfo  multiplying  the  number  A  B 

produce 


/ 


of  Euclides  Elementes, 


Fol6\. 


B 


\i% 


'F 


4* 


produce  the  number  £F:agayne  let  the  numberCi?  multiply¬ 
ing  the  felfe  fame  number  AB  produce  the  numberFG.  Now1 
foraftnuch  as  the  number^  C  multiplying  the  number  zA  B 
produced  the  number  EF:  therefore  the  number  zAB  meafu-* 
reth  the  number  E  F  by  the  vnities  which  are  in  A  C.Againe  for- 
afmuchasthe  number  CB  multiplied  the  number  ^4 £,andpro 
duced  the  number  F  G  =  therfore  the  number  A  B  meafureth  the 
number  FG  by  the  vnities  which  are  in  the  number  Ci? .But  the 
fame  number  AB  before  meafured  the  number  E  F  by  the  vni- 
tieswhich  are  in  the  number  AC.  Wherefore  the  number  A  B 
pieafureth  the  whole  number  S  G’by  the  vnities  whcih  are  in  A 
^.Farther  fo rafmuch  as  the  number  2?  multiplying  it  felfe  pro 
duced  the  number/?:  therefore  the  Humbert'S  meafureth  the  --,C  ^ 

number  D  by  the  vnities  which  are  in  himfelfe.Wherfore  it  mea  % jp 

fureth  either  of  thefe  numbersmamelyjthe  number  £>,{and  the 
number  E  G, by  the  vnities  which  are  in  himfelfe .  Wherfore  how 
multiplex  the  number  D  is  to  the  number  AB ,  fo  multiplex  is 
the  number  EG  to  the  fame  number  AB.  But  numbers  which 
are  equemultiplices  to  one  and  the  felfe  fame  number^re  equal 
the  one  to  the  other.  Wherefore  the  number  *D  is  equallto  the 
number  E  G .And  the  number  D  is  the  fquare  number  made  of  ^ 
the  n  urn  ber  A  £,and  the  number  E  G  is  compofed  of  the  two  fu- 
perficiall  numbers  produced  o£A B  into BC,  and  o£ B  Ainto  A 
C.  Wherefore  the  fquare  numberproduced  of  the  number  <A  B 
is  equall  to  the  fuperficial  numbers,produced  of  the  number  A  B  into  the  number  £  Cs 
and  ofAB  into  AC,added  together  .If  therefore  a  number  be  deuided  into  two  other 
number  s  &c,  which  was  required  to  be  proued. 


1 

D 


i 

B 


be  3.  The  or  erne,  The  3.  Tropofition . 

ff a  right  line  be  deuided  by  chaunceithe  reBanglefigure  com¬ 
prehended  vnder  the  whole  and  one  of  the  partes  ,is  equall  to 
the  reft  angle figu>  e  comprehended  vnder  the partes ,  &  "vnto 
the  fquare  which  is  made  of  the for ef aid part . 


Vppofe that  the right linegeuen  AB  be  deuided  bycbaunce  in  the 
point  C.Tben  1  fay  tbat  the  re  Handle  figure  compreheded  Vnder  tbe 
lines  A  &  and  B  C  is  equall  Vnto  tbe  re  Bangle  figure  comprehended 
Vnder  tbe  lines  A  C  and  C  B^and  alfo  Vnto  tbe  fquare  which  is  made 
of tbe  line  B  C.  fDefcribe(by  tbe  4.6, of  the  firfifvpon  tbe  line  B  Ca  fquare  CfD 
EB :  and  (by  tbe  fecond  peticion)extendE  Vnto  F.  And  by  tbe  poitit  A draw 
(by  tbe  3 1  .of  tbe  firft)a  line  parallel  Vnto  either  of  thefe  lines  C  B)  and  BE^and 
lettbe  fame  be  AFJSLow  tbe  paraUdo*  b  c  4 

grame  A  E is  equall  Vnto  the parallelo*  [~  1  — 

grammes  A  B)  and  C E,And  A  E  is  tbe 

re  Bangle  figure  comprehended  Vnder  |__ _ _ _ _ _ 

tbe  lines  A  B  and  B  C,For  it  is  compre*  E  b  p 

bended  Vnder  tbe  lines  A  B  and  B  E. 

%iif  but 


Conflmtlm, 


Demnftmio 


Thefecond'Booke 

"fohich  line  BE  if  e quail Vnto  the  line S Ct  jind the paradelograme  A D is  e* 
quail  to  that  which  is  contayned Vnder  the  lines  A C and  C  <B\  for  the  line  5)  C is 
equall Vnto  the  line  C  B .  And ID  B  is  the  fquare  Tbhich  is  made  of  the  lyne  C  B. 
VVherfore  the  re  Handle  figure  comprehended  Vnder  the  lynes  A  B  and  BC  is 
equall  to  the  re  Bangle  figure  comprehended  Vnder  the  lines  A  C  and  C  3  Ural* 
fo  Vnto  thefquare  Tbhicb  is  made  of  the  line  3  Ct  If  therfore  a  right  line  he  de* 
uided  by  chaunce%the  reBangle  figure  comprehended  Vnder  the  tohole  and  one 
of  the  partes fis  equall  to  the  reBangle  figure  comprehended  Vnder  the  parte  f, 
andvnto  the  fquare  ^hichis  made  of  the  forefayd  part:  H>hich  'to  as  required  to 
he  proued , 


An  example  ofthis  Propofition  in  numbers, 

Suppofe  a  number,namely,i4.to  be  deuided into  two  partes  8,and  6.  The  whole 
number  ^.multiplied  into  8.  one  of  his  partes,produceth  i  i2:the  partes  8. &<5.  mul¬ 
tiplied  the  one  into  the  other  produce  48,whicn  added  to  6^( which  is  the  fquare  of  8. 
the  former  part  of  the  number  Jamounteth  alfo  to  1 1 1 :  whiche  is  equall  to  the  former 
fumme.Asyou  fee  in  the  example. 


r  Multiplication  of  the  whole" 
into  one  of  his  partes. 

r  14 

8 

O: 

j-the  partes. 

■ 

II2_ 

— 

"the  number  produced  of  the-. 

whole  into  one  of  hispartes- 

Multiplication  of  the  one 

8 

part  into  the  other. 

' 

- 

6 

:>equa!  to 

- 

48 

48 

■ 

J± 

Multiplication  of  the  for¬ 
mer  part  into  it  felfe. 

8 

8 

112 

^the  number  compofed  of the 
one  partinto  the  other,  and- 
of  the  former  part  into  him- 
felfe. 

*4. 

J 


BarUam* 


The  demonftratton hereof folloveth  in  Barlaam, 

The  third  propofition. 

If  a  number geuen  be  deuided  into  tWo  numbers:  the fiperfciall  number  Which  is  produced  of 
the  multiplication  of ih  e  Whole  into  one  of  the  partes, ts  equall  to  the/uperficiall  number  which  it  pro - 
duced  of  the  partes  the  one  into  the  olher,andtotbe fquare  number  produced  of  the  forefayd  part. 


Suppofe  that  the  number  geuen  be  v4  #,which  let  be  deuided  into  two  numbers  A 
Cand  C  #  .Then /fay  that  the  fuperficiall  number  whiche  is  produced  of  the  multipli¬ 
cation  ofthe  number^.# into  the  number;#  C  is  equall  to  the  fuperficiall  number 
w'hich  is  produced  of  the  multiplication  of  the  number  A  C  into  the  number  C  #,and 
to  the  fquare  number  produced  of  the  number  C  '3. For  let  the  number  a A  B  muitipli- 
eng  thq.  number  C B  produce  the  number  D.And  let  the  number  A  C  multiplieng  the 
number  CB  produce  the  number  E .Ftand  finally  let  the  number  C B  multiplieng  him- 
felfe  produce  the  number  F  G.  Nowforafmuchas  the  number,//#  multiplieng  the 
'  -  number 


ofEuclides  Elements s. 


. 6 5* 


t 


4r 


A 


number  CB  produced  the  huitibefi)  .Therfore  the  number  C 
B  meafureth  the  number  D  by  the  vnities  whiche  are  in  the 
number  A  5.Agayne  forafmuch  as  the  number^  Cmultipli- 
ed  the  numberCi?,and  produced  the  number  E  F}  therefore 
the  number  C  B  meafureth  the  nuber  EF  by  the  vnities  which 
are  m  A C.Agayne  forafmuch  as  the  number  C B  multiplied  it 
felfe  and  produced  the  number  EG1-.  therfore  the  numberC# 
meafureth  the  number  F  G  by  the  vnities  which  are  in  it  felfe. 

But  as  we  haue  before  proued  the  felfc  fame  nuber  CB  mea¬ 
fureth  alfo  the  number  EF  by  the  vnities  which  ate  in  the  nu¬ 
ber  A  C,w herfore  the  number  CB  meafureth  the  whole  num¬ 
ber  EG  by  the  vnities  which  are  in  the  number  AB,  And  it  al¬ 
fo  meafureth  the  number  D  by  the  vnities  whiche  are  in  the 
number  A 2>  .Wherfore  the  number  CB  equally  meafureth  ei¬ 
ther  number,namely,the  number  ZJ^and  the  number  EG. But 
thofe  numbers  whomc  one  and  the  felfe  fame  number  mcafu- 
reth  equally,  are  equall  the  one  to  the  other.  Wherfore  the 
number  D  is  equall  to  the  number  E  G .But  the  number  D  is  a 
fuperficiall  number  produced  of  the  multiplication  of  the 
number  AB  into  the  number  2>C, and  thenumberACis  the 
fuperficial  number  produced  of  the  multiplication  of  the  nu¬ 
ber  AC  into  the  number  CB,  and  of  the  fquare  of  the  number 
CB. Wherfore  the  fuperficial  number  produced  of  the  multi¬ 
plication  of  the  number -^5  into  the  numbered  is  equal  to  the  fuperficiall  number 
produced  of  the  number  A  Cinto  the  number  CB,  ana  to  the  fquare  of  the  number  C 
.S.If  therfore  a  number  be  deuidedinto  two  numbers.the  fuperficiall  nuber  &c:  which 
was  required  to  be  proued. 


S 


1 

E 


\ The  ^Theorems.  The  4..  Tropofition , 

If  a  right  line  be  deuided by  chaunce,  the  fquare  whiche  is 
made  of  the  whole  line  is  equal  to  thefqmres  which  are  made 
of  the  partes,  &  vnto  that  rectangle figure  which  is  compre* 
bended  lender  the  partes  twife. 

B yppo/e  that  the  right  lyne  A  ©  he  hy  chaunce  deuided  in  the pointe  C. 
™|  Then  l Jay  that  the  fquare  made  of  the  line  A  ©  is  equall  Vntoy  [quarts 
which  are  made  of the  lines  A  CandC  B,  and  Vnto  the  re  Bangle  figure 
contained  Vnder  the  lines  A  C  and  C  ©  tftife.  Defer  the 
(by  y  46.0  f  the  first)  Vpon  the  line  AB  a  fquare  ADE 
©  land  draft  a  line  from  B  to  fD,and(by  the  3 1  .oft  he  h 
firft)by  the  point  C  draw  a  line  parallel  Vnto  either  of 
theje  lines  A  Bland  ©  E cutting  the  diameter  ©  X)  in 
the  point  G,and  let  the  fame  be  C  E.And(by  the  point 
G  (by  the  felfe  famefdraft  a  line  parallel  Vnto  eyther 
of the fe  lines  A  ©  and  BE,  and  let  the  fame  be  H 
And forafmuch  as  the  line  CF  is  a,  . parallel  Vnto 

S.L 


CwJlrH&ion • 


the 


Vemonfiratio 


(eft 

theline  AD,andvponthemfalletb  arigbtlineft  ft):  therfore(by  the  2  9, of  the 
firft)the  outward  angle  CG  ft  is  equallVnto  the  inward  and  oppojite  angle  A 
ft  ftJBut  the  angle  Aft  ft  is  (by  the  5*  of  the  fir  ft)  equali  Vnto  the  angle  A  ft 
ft:  for  the  fide  ft  A  is  equali  Vnto  the  Jide  A  ft  (by  the  definition  of  a  fquare ). 
Wherfore  the  angle  CG  ft  is  equali  Vnto  theangleG  ftC:  wherfore(  by  the  6, 
of  the  fir  ft  )t  he fide  ft  C  is  equali  Vnto  the  fide  C  G.ftut  C  ft  is  equallVnto  G 
and  C  G  is  equali Vnto  Kfft:  wberforeGJfjs  equallVnto  ft.  VVherfore  the 
figure  CGKfift  confifteth  of  four e  equali  ftdes.l  fay  alfo  that  it  is  a  reHangle  fi¬ 
gure. For  for  ajmacb  as  CG  is  a  parallel  Vnto  ftK&vpon  the  falleth  a  right  line 
Cft,therfore(byy  9. of the  i,)tbe  angles  JfJB  C^andG  C  ft  are  equal  vnto  two 
right  angles /But  the  angle  KfftC  is  a  right  angle  givher fore  y  angle  ftCGis  alfo 
a  right  angle VV her f  or e(by  the  3  4 yfthe  first )tbe angles  oppojite  Vnto  them, 
namely  yC  G  If,  and  G  Kjft  are  right  angles,  Wherfore  C  G  If  ft  is  a  re  Han* 
gle  figure.  And  it  Was  before  proued  that  the  fides  are  equali.  VVherfore  it  is  a 
fquare  ^andit  is  defcribedvpon  the  line  ft  C  And  by  the  fame  reafon  alfo  H  Fis 
a  fquare  ,and  is  defcribed  Vpon  the  line  H  Gjhat  isy>* 
p  on  the  line  A  C.VVberfore  the  f'quares  Ff  F  and  C  If 
are  made  of  t  he  lines  A  C  and  C  ft  And  forafmucb  as  H 
the  parade  lograme  AG  is  (by  the  45,  of  the  fir  ft)  e* 
quail  Vnto  the  parallelogramme  G  E.And  A  G  is  that 
which  is  contayned  V rider  AC  and  Cftjor  CGis  equal 
Vnto  C  ft,wherforeG  E  is  equali  to  that  which  is  con» 
tained  Vnder  A  C  and  C  ft,  V therefore  A  G  audG  E 
are  equali  Vnto  that  which  is  comprehended  Vnder  A 
CandC  ft  twife.  And  the  /quay  es  H  Eand  C  Jfaremade  of  the  lines  AC  and  C 
ft. VVherfore  theft  foure  reHanglefigures  HFflf^A  Go  and  G  E  are  equali 
Vnto  the Jquareswbiche  art  made  cf the  lines  AC  and  C  ft,andto  the  reH  angle 
figure  which  is  comprehended  Vnder  the  lines  AC  and  C  ft  twife.  ftut  the  reH * 
angle  figures  HF>  C  If,  A  G,and  G  Eare  the  whole  reHangle  figure  At)  Eft 
Which  is  the  fquare  made  of  the  line  A  ftJWherf ore  the  fquare  which  is  made 
of  the  line  Aft  is  equal!  to  the  fqmreswhicharemade  of  the  lines  A  CandC  fty 
and  Vnto  the  reHangle  figure  which  is  comprehendedvnder  the  lines  AC  and 
C  ft  t  wife. If  therfore  a  r  ight  line  be  deuided  by  chaunce ,  the  fquare  which  e  is 
made  of  the  whole  line  fis  equali  to  the  (quarts  which  are  made  of  the  partes^ 
Vnto  the  reHangle  figure  which  is  comprehended  Vnder  the  partes  twifeiwhich 
Was  required  to  be  proued * 


\{t- 


ration * 


:.u\ 


5'  v\ .  > 


I  fay  that  the  fquare  of  the  line  AftisequaH  Vrito  the  fquare  s  Wbiche  art 
made  of  the  lines  A  C  and  Cftjt?  vnto  the  reitangle  figure  which  is  compnbe* 
ded  Vnder  the  lines  AC  and  Cft  twife.Fpr the felfe fame  difcription  abiding for* 

5  afmuch 


B 


7 

/ 

/ 

ofEuclidcs  Elementes.  FoL  6  6 . 

a/much  as  the  line  AB  is  equal l  Vnto)  Ime  At>yy  angle  AB  t>  is(hy  the  of  the 
firH)  equal 7  Vnto  the  angle  A  T>  B. And  fora fmuch  as  the  three  angles  of  euery 
triangle  areequalto  two  right  angles  (by  the  5  2  of  the  fir  ft). therefore)  three 
angles  of  the  triangle  A  BS)ynamely^  the  angles  A  EDBfD  B  Afand  B  A  Dyare 
e  quail  to  two  right  angles.But  the  angle  B  AT>  is  a  right  angle  therefore  the 
angles  remayning  A  B  D>and  A  T)  By  are  equallvnto  one  right  angleiand  they 
are  equally  one  to  the  other ywher fore  either  of  thefe 
angles  A  B  Bdyzs'  A  Bfts  the  halfe  of  a  right  angle, 

And  the  angle  BCG  is  a  right  angle ,  for  it  is  equall 
Vnto  the  oppofite  angle  at  the  point  A  (by  the  2  9.  of 
the  fir  ft). V therefore  the  angle  rtmaymng  CG  Bis 
the  halfe  of  a  right  angle.  Wherefore  the  angle  CGB 
is  equal Vnto  theangle  CBG:  therefore  alfo  the  Jide 
BC is  equallvnto  thefide  CG.But  BC is  equall  Vnto  u 
G  and  C  G  is  equal  Vnto  B  Wherefore  the  fin 

gureC  K^confiftetb  of  equall fidesiand  in  it  is  a  right  angle  C  Bl<f.  Where » 
fore  C  Kjs  a  fquare, and  is  made  of  the  line  B  C.  And  by  the  fame  reafon  H  F 
is  a fquare,and  is  equall  Vnto  that fquare  which  is  made  of  the  line  A  Q  Where* 
fore  C  Kjnd  El  Fare  fquares^and  are  equall  to  thofefquares  which  are  made 
of  the  lines  A  Cand  C  B.  Andforafmuck  as  AG  is  equall  Vnto  E  G:and  AG  is 
that  which  is  contaynedvnder  A  Cand  C  BJorGC  is  equal  Vnto  C  Biwhere* 
fore  E  G  alfo  is  equall  to  that  which  is  coprehended  Vnder  A  C  and  C  B:  where* 
fore  AG  and  EG  are  equall  Vnto  that  r  eBangle figure  which  is  comprehended 
Vnder  A  Cy  and  C  B  twife.AndCK^and  H  Fare  equal  Vnto  the f quires  which 
are  made  of  A  Q  and  (fB:  wherefore  C^HFjAG,  and  GE  are  equal  Vn - 
to  thofefquares  which  are  made  of  A  C,and  CBjnd  Vnto  that  re  El  angle  figure 
Which  is  comprehended  Vnder  A  Cand  C  B  twife.  But  C  Jf^ElFA  GyandG  E 
are  the  whole  fquare  A  E  which  is  made  of  A  B ,  Wherefore  the  fquare  which 
is  made  of A  Bfts  equall  to  the  fquares  which  are  made  of  AC  and  C  By  and  Vn* 
to  t  he  r  eft  angle  figure  which  is  comprehended  Vnder  A  C  and  CB  tWife:  which 
was  required  to  be  demonstrated. 


Hereby  it  ismanifeft  that  the  farallelogrames  Vebich  conftfl  about  the  diameter  of  a fan  art  mtfi  ^  Corollary* 
needes  be /quarts. 

This  proportion  is  of  infinite  vfe  chiefcly  infurde  numbers.By  helpeofitis 
made  in  theadditio  Sc  fubftra&ion,alfo multi plicatio  in  Binomials  &  refidu- 
als.  And  by  hclpc  hereofalfo  is  demonftratcd  that  kinde  of  equation,  which  is 
when  there  are  three  denominations  in  naturail  order,  or  equally  diftant,  and 
two  of  the  greater  denominations  are  equall  to  the  thirde  being  lefie  Onthis 
propofition  is  grounded  theextraftion  of  fquare  roots.And  many  other  things 
are  alfo  by  it  demonftrated* 

s.tj. 


An 


Balaam, 


fe  The  fecondBookg 

An  example  of  this  Tropofition  in  numbers . 

Suppofe  a  number  namely,  x  7. to  be  deuided  into  two  partes  9.  and  8.  The  whole 
number  1 7  .multiplied  into  him  felfe,produceth  289. The  fquare  numbers  of  9.  and  8. 
are  8 1.  and  64:  the  numbers  produced  of  the  multiplication  of  the  partes  the  one  in- 
to  the  other  twife  are  72,  and  72:  which  two  numbers  added  to  the  fquare  numbers 
°t  9,and  8.namely,to  81.  and  6 4.  makealfo  3  8p.whichiscquall  to  the  fquare  number 
of  the  whole  number  1 7 .  As  you  fee  in  the  example. 


The  multiplication  of  the-', 
whole  into  himfelfe. 

r 17 

17 

{%: 

>-the  partes  of  the  whole 

n  9 

17 

28p 

r  the  number  produced  of  the 

whole  into  himfelfe. 

The  multiplication  ofeche 

9 

part  into  himfelfe. 

9 

81 

^equall  to 

8 

81 

8 

*4 

<54 

7a 

7* 

-  .  •  .  .V.  ‘ 

289 

"•the  number  compofed  of  cche 

The  multiplication  of  the 
one  part  into  the  other 

9 

8 

part  into  himfelfe.andof  the  one 
into  the  other  twife. 

twife. 

72 

-  -  . . ,  ..  J  A  * 

9 

8 

- 

L  7» 

The  demonftration  wherof  followcth  inBarlaam, 


The  fourth  Tropofition . 

If  a  number  geuen  be  deuided  into  tWo  numbers :  thejejuare  number  ofthewbole,is  c/juatt  to  the 
fcjuare  numbers  of  the  partes,  and  to  thefuperficiall  number  which  is  produced  of  the  multiplication 
of  the  partes  the  one  into  the  other  tWtfe. 

Suppofe  that  the  number  geuen  bet  AH:  which  let  be  deuided  into  two  numbers 
A  C  and  CB.  Then  I  fay  that  the  fquare  number  of  the  whole  number  -/42?,is  equall 
to  the  fquares  of  the  partes,that  is,to  the  fquares  of  the  numbers  C  and  CA,and  to 
the  fuperfkiali  n  umber  produced  of  the  multiplication  of  the  numbers  AC  and  CB  the 
one  into  the  other  twife.  Let  the  fquare  number  produced  of  the  multiplication  of 
the  whole  number  AB  into  himfelfe  be  D .  And  let  C  A  multiplied  into  himfelfe 
produce  the  number£-F;  KndfB  multiplyed  into  it  felfe  let  it  produce  GH:  andfi- 
nallyofthemultiplicatioofthe  numbers  A  CandCAthe  one  into  the  other  twife  let 
there  be  produced  either  of  thefe  fuperfkiali  numbers  F  G  and  Ft  K .  Now  forafmuche 
as  the  number  e^/Cmultiplyingit  felfproduced  the  number  .E  A;  therefore  the  num¬ 
ber  zAC  meafureth  the  number  EF  by  the  vnities  which  are  in  it  felfe.  Andforafmuch 
as  the  number  CB  multiplyed  the  number  C  A  and  produced  the  number  F  G :  there- 


ofEuclides  Elementes «  Fc-Ldj. 

fore  the  number  ^Cmeafureth  the  nuber  F  G  by  the  vnities 
whiche  are  in  the  number  CB.  But  it  before  alfo  meafured 
the  number  SFby  the  vnities  which  arein  it  felfe.  Where¬ 
fore  the  number  A  B  multiplying  the  number  -4Cprodu- 
ceth  the  number  E  G\And  therefore  the  number  EG  is  the 
fuperficiall  number  produced  of  the  multiplication  of  the 
number  B  A  into  the  number  C,  And  by  the  fame  rea- 
fonmay  weproue  that' the  number  G K  is  the  fuperficiall 
number  produced  of  the  multiplication  of  the  number  A  B 
into  the  number  #C.Farther  the  number  Z> is  thefquare  of 
the  number  A  B.  But  if  a  number  be  deuided  into  two  num¬ 
bers,  the  fquare  of  the  whole  number  is  equall  to  the  two 
fuperficiall  numbers  which  are  produced  of  the  multipli¬ 
cation  of  the  whole  into  either  the  partes  ("by  the  2«Theo- 
reme.)  Wherefore  the  fquare  number  D  is  equall  to  the  fu¬ 
perficiall  number  E  K.  But  the  number  EKis  coinpofed  of 
the  fquares  of  the  numbers  A  C  and  C  B,  and  of  the  Superfi¬ 
cial  number  which  is  produced  of  the  multiplication  of  the 
nuber  A  C and  C B  the  one  into  the  other  twife:&  the  num¬ 
ber  D  is  the  fquare  of  the  whole  number  AB.  Wherfore  the 
fquare  nu  mber  produced  of  the  multiplication  of  the  num¬ 
ber  A  B  into  himfelfe,  is  equall  to  thefquare  numbers  of 
the  partes,  that  is,to  the  fquare  numbers  of  the  nuber  sAC 
and  C2?,and  to  the  fuperficiall  number  produced  of  the  multiplication  of  the  num¬ 
bers  A  C  and  C  B ,  the  one  into  the  other  twife.  If  therefore  a  number  geuen  be  deui¬ 
ded  into  two  numbers  &c  .Which  was  required  to  beproued. 

The  5.  T beoreme .  The  jfPropofition, 

%  If  a  right  line  he  deuided into  two  equall partes^  &  into  two 

ynequall partes:  the  reUangle  figures  comprehended  ynder 
the  ynequaH partes  of  the  whole ^together  with  the fquate  of 
that  which  is  betwene  the  feUiosJs  equal  to  the  fquare  which 
h  made  of  the  halfe. 

mqyppof6  that  the  right  line  A  B  be  derided  i  nto  tWo  equall  partes  in  the 
Cydnd  into  two  wequall  partes  in  the  point  D.Tben  I  fay  that  the 
reSlangle  figure  comprehended  Vnder  AD  and  D  B  together  With  the 
fquare  which  is  made  ofC  Dy  is  equall  to  thefquare  which  is  made  ofC  B,  De*  ConftrutHon . 
Jcribe(by  the  4 6.  of  the  firfi) 

VpponCB  a  fquare y  and  let  the 
Jame  be  C  EFB,  jfnd(hy  the 
firfi  peticion)draWe  a  line  from 
E  to  B.And  byy point  Ddrawe 
(by  the  $  \tof the  firfi )  a  line  pa • 
rallelvnto echo/ the fe  lines  CE 
and  B  E  cutting  the  diameter  B 
E  in  the  point  H}  and  ktey  fame 
be  ID  C.  And  agayne(by  the  felfe 


Satj. 


6-f 


’  L 

41 

b 

\x 


-H 

■  F 


b-  A** 


3* 


A  V  E, 


fame 


Dmenfiratio 


a 


TheJiconcNSookg 

fame)  by  the  point  H  dr aWe  aline  parallel  Vnto  ecbe  oftbefe  lines  A  B  and 
EFyand  let  the  fame  be  KJ)  \  and  let  Kj)  be  equall  'Vnto  AB,  Andagaine 
(by  the  ( elfe Jame  )by  the  point  A  draw  a  line  parallel  Vnto  either  of  thefe  lines 
C L  and  B  0,  and  let  the  fame  be  AI\.  Andforafmuch  as  (by  the  43 ,of  the 
firfl )tbe  f  upplement  CH is  equall  to  the  fupplemet  HF.put  the  figure  ID  0  co* 
mon  Vnto  them  both .  VF'herefore  the  whole  figure  CO  is  equall  to  the  whole  fi * 
gure  DF.But  the  figure  CO  is  equallvnto  the  figure  ALt  for  y  line  AC  is  equall 
Vnto  the  line  C  B.  Wherefore  the  figure  AL  alfo  is  equal  Vnto  the  figure  ID  F. 
But  the  figure  C  H  common  Vnto  them  both  J/V her fore  the  whole  figure  AH. 
is  equall  Vnto  the  figures  D  L  and  D  F.  But  A  H  is  equall  to  that  which  is  co* 
tayned  Vnder  the  lines  A  D  and  D  Bfor  D  His  equall  Vnto  D  B.And  the  fi¬ 
gures  F  D  &DL  are  the  Gno - 
tnon  MKXynerfore  y  Gno*  A 
mon  MNX  is  equall  to  that 
Which  is  contayned  Vnder  A  D 
and  V  BjPut  the  figure  LG  co • 
mon  Vnto  them  both^which  is  e* 
qual  to  the fquare  Which  is  made 
of  C  (D,  VFherefore  the  Gnomo 
M  2^1  X  and  the  figure  L  G  art 
equall  to  the  rectangle  figure  co - 

prehended  Vnder  X  D  and  D  B  and  Vn(o  the  j quart  which  is  made  ofODjBut 
t  he  Gnomon  M  TL  Xfand  the  figure  L  G  are  the  whole  fquare  C  E  FB%whick 
is  made  ofBC.VF  here  fort  the  reft  angle  figure  comprehended  Vnder  A  ID  and 
1)  B,  together  with  the  fquare  which  is  made  of  CD yis  equall  to  the  fquare 
Which  is  made  ofCB,  If  therefore  a  right  line  be  deuided  into  two  equall parts  > 
and  into  two  Vnequall partes  yhe  re  ftangle  figure  comprehended  Vnder  the  Vn» 
equall  partes  of  the  whole  ^together  with  the  fquare  of  that  which  is  betWene 
the  /eft  tons,  is  equall  to  the  fquare  which  is  made  of  the  halfe:  which  Was  requia 
redtobeproued , 

This  Propofitionalfoisofgrcate  vfein  Algebra.  By  it  is  demonftrated 
that  equation  wherein  the  grcatell  and  IcaitWe&es  or  numbers  are  equall  tj 
the  middle. 


r 

M 

K 

L 

M  / 

H  / 

AS 

P 


An  example  of  this  propofition  in  numbers. 

Take  any  number  as  20:  and  deuide  itinto  two  equall  partes  xo-and  10.  and  then 
into  two  vnequall  partes  as  15.  and  7.  And  take  the  differcce  of  the  halfe  to  one  of  the 
vnequall  partes  which  is  5.  And  multiplythevnequallpartcs,thatis,i?  andy.theone 
into  the  other,which  make  pi.take  alfo  the  fquare  of  3  . which  is  p.  and  adde  itto  the 
forefaydenumberpuandfolhallthercbe  made  xoo.  Then  multiply  the  halfe  of  the 
whole  number  into  himfelf,  that  is,  take  the  fquare  of  10. which  is  100. which  is  equal 
to  the  number  before  produced  of  the  multiplication  of  the  vncqual  parts  the  one  in¬ 
to  the  othcr,&  of  the  difference  into  it  felfe  which  is  alfo  1  oo,As  you  fe  in  the  example* 

The 


of Euctides  Elmtntes* 

The  whole  euefl  ^  ao  fio 


iuy» 


Multiplication  of  the  vn¬ 
equall  partes  the  one  into 
the  other* 


Multiplication  of  the  dif¬ 
ference  into  it  felfe. 


Multiplication  of  the  half 
into  it  felfe* 


io 

xo 


1  ^  ^-the  vnequall  partes 


w  3  f^The  difference  of  ohe  of 
the  vnequal  partes  to  the 
halfe. 


X 


*1 

7 

9*  Pl 

9 


j  iOo 

3 


io 

io 


too 


'  the  huber  coiiipofed  of  the  mul¬ 
tiplication  of  the  vnequal  partes 
the  one  into  the  other*  &  of  the 
difference  into  it  felfe 

■cquall  to 


H  X 


the  number  produced  of  the 
halfe  into  it  felfe. 

The  demonftration  'wher°ffDllowech  in  Barlaam, 

The  fifth  proportion. 

If  an  earn  number  be  derided  into  Wo  e  quail  partes^and  againealfo  into  Wo  vnequall  partes: 
the fuperfit'iull  number  Which  is  produced  of  the  multiplication  of  the  vnequall  partet  the  one  into1 
the  other  together  with  the  fquare  of  the  numberfet  betwene  the  parts ,  is  equal  to  the  fquare  efhalft 
the  number. 

■Suppofc  that^Abeaneuen  number :  which  let  be 
deuidfcd  into  two  equal!  numbers  A  Cand  CBy and  into 
two  vnequall  numbers  A  Jf  &nd  D  B.  Then  7  fay, that  the 
fquare  number  which  is  produced  of  the  multiplication 
Ofthe  halfe  number  CJ3  into  it  felfe  Js  equal!  to  thefu- 
perficiall  number  produced  of  the  multiplication  of  the 
vnequall  numbers  A I)  and  D  B  the  one  into  the  other, 
and  to  the  fquare  number  produced  of  the  number  C  D 
which  is  fet  betwene  the  fay de  vnequall  partes.  Let  the 
fquare  number  produced  of  the  multiplication  of  thb 
halfe  number  CB  into  it  felfe  be  E .  And  let  the  fuperfU 
ciall  number  produced  of  the  multiplication  of  the  vne-, 
qual  nuber  s  A  D  and  D  B  the  one  into  the  other,be  the 
number  FCj-.  and  let  the  fquare  of  the  number  DC  which 
is  fet  betwene  the  partes  be  G  H,  Now  forafmueh  as  the 
number  B  Cisdeuided  into  the  numbers  BD  and  DCj 
therforethe  fquare  of  the  number  2?  C,that  is,  the  num¬ 
ber  E,is  eq  uall  to  the  fquares  of  the  num  bers  B  D  and  t) 

C3andto  the  fuperficiall  number  which  is  compofed  of 
the  multiplication  of  the  numbers  B  D  and  t>  C  the  one 
into  the  other  twife,f  by  the 4.propofition  of  this  boke) 

Let  the  fquare  of  the  number  E  D  be  the  number  K  L:& 
let  N  X  be  the  fquare  of  the  number  D  C:  and  finally  of 

S.iiii.  the 


3 


b  -.'D 


4 


\  6" 


b  *  C 


• 

4 

;  i  ■ 

4 

Ou 

A* 

M.H 

%% 

4 

-f 

4 

i. 

n 


m 


E 


the  multiplication  of  the  numbers  BD  and  D  C  the  one  into  the  other  t\vife,let  be  pro 
duced  either  of  thefe  number's  L  M  and  M  2^.  Wherefore  the  whole  number  K X  i$ 
equall  to  the  number  £  Andforafmuch  as  the  number  BD  multipliyngit  felfe produ¬ 
ced  the  number  K L, therefor  it  meafureth  it  by  the  vnities  which  are  in  it  feife«  More 
ouer  forafmuch  as  the  number  CD  multiplying  the  number  B  D  produced  the  num¬ 
ber  L  A -/,  therefore  alfo  D  B  meafureth  L  M  by  the  vnities  which  are  in  the  number  C 
Z>:  but  it  before  meafured  the  number  K  L  by  the  vnities  which  are  in  it  felfe,  Where- 
fore  the  number  D  B  meafureth  the  whole  number  KM  by  the  vnities  which  are  inC 
B .  But  the  number  CB  is  equal!  to  the  number  C  A.  Wherefore  the  number  'DB  mea¬ 
fureth  the  number  K  A4  by  the  vnities  which  are  in  C  A.  Agayne  forafmuch  as  the  nu- 
ber  CT)  multiplivng  the  number  D  B  produced  the  number  A/ 2W therefore  the  num¬ 
ber  D  B  meafureth  the  number  M  7^ by  the  vnities  which  are  in  the  number  CD -but 
it  b  efore  meafured  the  number  KM  by  the  vnities  which  are  in  the  number  ^C.Wher 
fore  the  number  B  D  meafureth  the  whole  number  KN  by  the  vnities  which  are  in  the 
numbers  D.  Wherefore  the  number  FG  is  equall  to  the  number  if  iV.For  numbers 
which  are  cquemultiplices  to  one  and  the  felfe  fame  number,are  equall  the  one  to  the 
other.ButthenumberG’i/ is  equall  to  thenumbcrNX:  foreither  ofthem  is  fuppo- 
fed  to  be  the  fquare  of  the  number  C  D.’Wherefore  the  whole  number  KX  is  equall  to 
the  whole  number  F  H, But  the  number  K  X  is  equall  to  the  number  E.  Wherefore  alfo 
the  number  F  His  equall  to  the  number  £.And  the  number  F His  the  fuperficial  num¬ 
ber  produced  of  the  multiplication  of  the  numbers  and  DB  the  one  into  the  o- 
ther  together  with  the  fquare  of  the  number  DC,  And  thenumber  £  is  the  fquare  of  the 
nutnberC£.Wherfore  the  fuperficiall  number  produced  of  the  multiplication  of  the 
vnequal  partes  AD  and  DB  the  one  into  the  other,  togetherwith  the  fquare  of  the  nu- 
ber  D  C  which  is  fet  betwene  thofe  vnequall  partes, is  equall  to  the  fquareof  the  num¬ 
ber  C  2?,  which  is  the  halfe  of  the  whole  number  AB.  If  therfore  an  euen  number  be  de- 
uidedinto  two  equall  partesAx*  which  was  required  to  beproued. 


1  I 


The  6fTheoreme,  The  6,cPropofition. 


'  -  •  .  3  -  ' 

If  a  right  line  be  deuided  into  typo  equal partes, and ifvnto  it 
be  added  an  other  right  line  direBly,the  reB angle  figure  con* 
tayned  vnder  the  whole  line  with  that  which  is  added,&the 


line  which  is  addedt  together  With  the  fquare  which  is  made 
of  the  halfe ,  is  equall  to  the  fquare  which  is  made  of  the  halfe 
line  and  of  that  which  is  added  as  of one  li  ne. 

'■  '  !  •  •  in  •  f-j'  is  ■<  C1.  Hi 


Vppcfe  that  the  rigbte 
line  A’Bbe  deuided  in - 
to  two  equall  partes  in 
point  C :  O'  let  there 
added  Vnto  it  an  other  right  line 
J)rBdirecHy}that  is  tojayynhich 
being  ioyned  Vnto  A  make  both 
one  right  line  A  Then  I fay, 
that  the  rectangle  figure  compre 
bended  )mder  A®  and 


'  •  ..  i..!i 

.'mSfcsih. 


ofEuclides  Elementes.  FoL  6p. 

gether  "frith  the  fquare  whiche  is  made  of  BC  is  equall  to  the  fquare  wU'chefit 
made  ofD  C.DeJcribe(  by  the  46  .of the  i  ,)vpon  CD  a  fquare  C  E  FD,and  (by 
the  firfi  petition  )draW  aline  from  D  to  E:and  (by  the  tuoftbe  first)  byy  point 
B  draw  a  line  parallel  Vnto  either  of thefe  lines  EC&*DFy  cutting  the  diame * 
ter  D  Em  the  point  H^and  let  the  fame  be  BG,&(by  the  felffitme  )by  y  point 
EL  draw  to  either  of  thefe  Itnes  A  T)  andEFa  parallel  line  M:  and  moreouef 

by  the  point  e  A  drawe  a  line  pa a 
rallelto  either  of  thefe  lines  CL 
and  D  M:  and  let  the  fame  be  A 
JfAnd  forafmuch  as  AC  is  e  quail 
Vnto  C B^therfore  (by  the  5  6,  of 
the  firfi)  the  figure  A  Lis  equal 
Vnto  the  figure  C  EL  But(by  the 
43  .of the  firfi  )C  His  equal  Vn* 
the  figure  H  F,  "frherfore  A  L  is 
equall  Vnto  FL  F.  Tut  the  figure 
C  M common  to  them  both^wber 
fore  the  whole  line  A  Mis  equal 
Vnto  the  gnomon  N  X  OjBut  A 
Mis  that  which  is  contaynedvnder  AT)  andD  B:  for  D  M  is  equal  Vnto  TfBi 
‘ frherfore  the  gnomon  TsL  %  OJs  equall  Vnto  the  reBangle  figure  contained  Vn* 
der  A D  and  D  BJPut  the  figure  L  O  common  to  them  both/frhich  is  equall  to 
the  fquare  "frhich  is  made  ofC  B ,  Wherefore  the  reBangle  figure  which  is  con* 
taynedvnder  AD  and  D  B  together  with  the  fquare  which  is  made  ofC  Bis  e* 
quail  to  the  gnomon  IflCO^and  Vnto  L  Gt  But  the  gnomon  NXO  and  L  G 
are  the  whole  fquare  CE  ED  which  is  made  ofC  D4  V fiber  fore  the  reBangle 
figure  contaynedvnder  A  D  and  D  B  together  with  the  fquare  which  is  made 
ofC  B  is  equall  to  the  fquare  which  is  made  ofC  D.Iftherforea  right  line  be  de* 
aided  into  two  equall  partes ,  and  Vnto  it  be  added  an  other  right  line  direBlyi 
the  reBangle  figure  contayned  Vnder  the  whole  line  with  that  which  is  added9 
and  the  line  which  is  added  ^together  with  the  fquare  which  is  made  of  the  halfe , 
is  equall  to  the  fquare  which  is  made  of  the  halfe  line  and  of  that  which  is  added 
as  of  one  line :  which  Was  required  to  be  demonstrated. 

By  this  Propofition(be Tides  many  other  vfcs)  is  in  Algebra  dcmonftrated 
that  equation  wherin  the  two  leffe  numbers  be  equall  to  the  number  of  the  grea^ 
tell  denomination* 

An  example  of  this propofition  in  number s4 

Take  any  euen  number  as  1 8  .and  adde  vnto  it  any  other  number  as  3  .which  mafe 
in  all  2  1  .And  multiply  2 1 ,  into  the  number  added,  namely,  into  3  ♦  which  maketh  6 3  . 
Take  alfo  the  halfe  of  the  wholeeuennumber,thatis,of  i8.whichis  p.  And  multiply  p. 
into  it  felf  which  maketh  8 1  .which  adde  vnto  63  .(the  number  produced  of  the  whole 
cuen  number  jand  the  number  added  into  the  number  added)and  you  fhal  make  144., 


AC  3  i> 


ThefecondTlooke 


Then  a  dde  p.thehalfe  of  the  whole  euen  number  vnto  3  .the  number  added  which  ma- 
keth  12  .And  multiply  i2.intoitfeIfe,thatis,take  the  fquare  of  12, which  is  144.WWCI3 
is  equall  to  the  number  compofed  of  the  multiplication  of  the  whole  number  and  the 
number  added  into  the  nu  m  ber  added,and  of  the  fquare  of  the  number  added ,  which 
is  alfo  144.  As  you  fee  in  the  example. 


’•The  whole  euen  number.-^ 
The  number  added. 


The  halfe  of  the  whole. 
The  number  added. 


2 1  the  number  compofed  of  the  whole  number^ 
the  number  added. 

9 

3 


Multiplication  of the 
whole  &  the  number  ad- 
dedintothe  number  ad¬ 
ded. 


12  the  number  compofed  of  the  halfe  andof  thf 
number  added. 

2 1 
3 


63 


*3 

81 


Multiplicand  of  thehalfe 
into  it  felfe. 


Multiplicand  of  the  halfe 
and  the  number  addedin 
to  it  felfe. 


9 

9 


x44 


81 

it 

12 

24 

12 

Lx44 


■'■the  nuber compofed  ofthe  whole- 
and  the  number  added  into  the 
number  added  and  of  the  fquare 
of  the  halfe 

■equall  to 


'"the  fquare  number  made  ofthe-’ 
number  compofed  of  the  halfe  and 
the  number  added. 


The  demonftration  wherof  folio  weth  in  Barlaam* 

The  fix t  Tropofition. 

If  an  euen  number  be  deiiided  into  two  equall  numbers, and -vnto  it  be  added jbme  other nttmheri 
the fuperfcia/l  number  Which  is  made  ofthe  multiplication  ofthe  number  compofed  ofthe  whole  num¬ 
ber  and  the  number  added,  into  the  number  added 5  together  With  the fquare  ofthe  halfe  number,  is 
equall  to  the fquare  of  the  number  compofed  ofthe  halfe  and  the  number  added. 


Suppofethat^2?bean  euennumber.andletitbedeuidedinto  two  equall  num¬ 
bers  AC  and  CB :  and  vnto  it  let  there  be  added  an  other  number  B  D.  Then  /fay  that 
the  fuperficiall  number  produced  ofthe  multiplication  ofthe  number  A'T>  into  the 
number  D/?  is  equall  to  the  fquare  ofthe  number  CD,  For  let  the  fquare  number  of 
the  number  C  D  be  the  number  £,and  let  the  fuperficial  number  produced  of  the  mul¬ 
tiplication  of  the  number  A  D  into  the  number  D  B  be  the  number  F  C.-and  finally  let 
the  fquare  number  of  CB  be  the  number  G  H.  And  forafmuch  as  the  fquare  of  the  nu- 
ber  CD  is  (by  the  4.propofition  Jequall  to  the  fquares  of  the  numbers  D  B  and  BC  to¬ 
gether  with  the  fuperficiall  number  which  is  produced  ofthe  multiplication  ofthe 
numbers  DB  and  BC  the  one  into  the  other  twife.  Let  the  fquare  of  the  number  BD 
be  the  number  KL:  and  let  the  fuperficiall  numbers  produced  ofthe  multiplication 
ofthe  numbers  D  B  and  2?C the  one  into  the  other  twifebe  either  of  thefe  numbers 

LM 


ofEuclides  Elementes.  FoL  jot 

LM  and  M  N;  and  finally  let  the  fquare  of  the  number  2?  Cbe  the  number  NX.  Wher- 
fore  the  whole  number  K-Afhall  be  equall  to  the  fquare  of  the  number  C  D.  But  the 
fquare  of  the  number  C  D  is  the  number  E.  Wherfore  the  number  K  X  is  equall  to  the 
number  F.  And  forafinuch  as  the  number  B  D  multiplieng  it  felfe  produced  the  num¬ 
ber  K  L:  therfore  the  number  B  D  meafureth  the  num¬ 
ber  K  L7  by  the  vnities  which  are  in  it  felfe,  but  it  alfo 
meafureth  the  number  L  M  by  the  vnities  which  are  in 
the  number  CB, Wherfore  the  number  D  B  meafureth 
the  whole  number  K  M  by  the  vnities  which  are  in  the 
number  C  D.  The  nuber  D  B  alfo  meafureth  the  num¬ 
ber  M  N  by  the  vnities  which  are  in  the  number  C B:  & 
the  number  CB is  equall  to  the  number  C  A  by  fuppo- 
fition.  Wherfore  the  number-D  2?  meafureth  the  whole 
number  K  N  by  the  vnities  which  are  in  the  number  A 
J>.Butthenumber2)2?  doth  alfo  meafure  the  number 
F  (7  by  the  vnities  which  are  in  the  number  AD  :  for 
by  fuppofition  the  number  F  Cj  is  the  fuperficiall  num¬ 
ber  produced  of  the  multiplication  of  the  numbers  A 
D  and  -OF  the  one  into  the  other*  Wherfore  the  num¬ 
ber  FCis  equall  to  the  number  KN.  But  the  number 
HG  is  equall  to  the  number  N  X :  for  either  of  them  is 
thefquare  number  of the  number  CB,  Wherefore  the 
whole  number i7  His  equall  to  the  number  KX:  and 
the  number  KX  is  proued  to  be  equall  to  the  number 
F.W herfore  the  number  F  f/fhall  alfo  be  equal!  to  the 
number  E,  And  the  number  F  A?  is  the  fuperficiall  num 
her  produced  of  the  multiplication  of  the  numbers^ 

%)  and  DB  the  one  into  the  other,together  wyth  the 
fquare  of  the  number  CBi  and  the  number  Fis  the  fquare  of  the  number  CD ,  Wher¬ 
fore  the  fuperficiall  number  produced  of  the  multiplication  of  the  numbers  a/f  D  and 
J>  B  the  one  into  the  other,  together  with  the  fquare  ofthe  number  CB ^  is  equal!  to  the 
fquare  of  the  number  C  D. 2f therfore  an  euen  number  &c* 

The  jJTheoreme ,  The  yfPropofition , 

ffa  right  lyne  be  deuided by  chaimce,  the  fquare  whicbe  is 
.  made  of  the  whole  together  with  the  fquare  which  is  made  of 
one  ofthe partes  Js  equall  to  the  rectangle  figure  which  is  co<* 
tayned  vncler  the  whole  and  the  [aid parte  twifie 3  and  to  the 
fquare  which  is  made  ofthe  other part 

Fppof'e  that  the  right  line  A  B  he  deuided  by  chaunce  in  the  point  C, 
Then  I fay  that  thefquare  Which  is  made  of  A  together  With  the 

fquare  which  is  made  ofB  Cfs  equal  Vnto  the  r  eh  angle' figure  which 
is  contained Vnder  the  lines  ABandBC  twifeyand  Vnio  the  fquare 
‘ Which  is  made  ofj.  C/Dejcribe(  by  the  46  .ofthe firfi)  Vppon  A  B  a  fquare  A 
&)E  B>arid  make  complete  the  figure  .And forafinucb  as  (by  the  45, ofthe  fir [l ) 
ihe  figure  A  G  is  equall  vnto  the  figure  G  EfFut  the  figure  C  F  common  to  the 

T,  '  both] 


v 


3 


c 


b*  *1. 


r 

r 

>  * 

- ♦ 

_ Xjf 

c . 

9  1 

49 

_  r 

1 

O 

« 

H 


i/A 


2 r 


I  E  F  K 


*1  he fecond'B  coke 

both: Ibher fore  the  'tobole  figure  A  Fis  equall  to  the  Tbhole  figure  C  B.  VVher* 
fore  the  figures  AFatid  C  E  are  double  to  the  figure  A  F.But  the figures  jfF 
and  CE  are  the  gnomon  Ffh  M,and  the fquare  CF:  "tober fore  the  gnomon 
L  M,and  the  fquare  C  F  is  double  to  the  figure 
AF.  But  the  double  to  A  Fis  that  Tbhicb  is  con* 
tayned  Vnder  ABandBC  ffrife,  for  BE  is  e* 
quail  Vnto  B  C  W  herfore  the  gnomon  FfL  M 
and  the  f quart  C  F  is  equal!  Vnto  the  rebiangle 
figure  contayned  Vnder  A  B  and  B  C  twifefPut 
the  figure  DG  commonVnto  them  both ,  "tohich 
is  thefquare  made  of yi Ct  Wberfore  thegno* 
mon  l^L  M  and  the  fquares  B  G  and  G  ID  are 
equal  Vnto  the  rebiangle  figure  which  is  contain 
ned  Vnder  AB  &  BCtwife9(sr  Vntothe fquare 
which  is  made  of  A  C.But  the  gnomon  the fquares  BG%<sr  DG  are y 

“tobole  fquare  BAD  Ey(sry  part  or  fquare  C  F9which fquares  are  made  of  the 
lines  AB  &  of  BCyherf ore  y fquares  which  are  made  of  A  B  O' B  Care  equal i 
Vnto  the  re  biangle  figure  which  is  contayned  Vnder  ABandB  Ctwife9andalfa 
Vnto  the  fquare  of  A  C.Iftherfore  a  right  line  be  deuided  by  chaunceithe fquare 
whichis  made  of  the  whole  together  with  thefquare  which  is  made  of  one  of  the 
partes, is  equall to  the  reblangle  figure  which  is  contayned  Vnder  the  whole  and 
the  f ay  d  part  twi[e9and  to  the  fquare  Which  is  made  of the  other  parte :  whiche 
Was  required  to  be  demonstrated . 


Fluflates  addeth  vnto  this  Proposition  this  Corollary* 

The [quarts  of Wo  vn  equall  lines  do  exc  cede  the  reft  angle  figures  contayned  vnder  the fitid  Unit 
by  the fquare  of  the  exceffe  voherby  the greater  lyne  excedeth  thclejfe. 

For  if  the  line  oA  B  be  the  greatcr,and  the  linci?  C  the  lefle.it  is  manifeft  that  the 
fquares  of  AB  andi?  C  are  equall  to  the  redangle  figure  contayned  vnder  the  lyncs  A 
B  and  rB  C twife,  and  moreouer  to  the  fquare  of  the  line  A  Cwherby  the  line  A  B  cxcc- 
derh  the  line  BC, 

By  this  propofition  moft  \ronderfully  vas  found  out  the  extra&ion  of  roote 
fquares  in  irrationall  numbers,befide  many  other  ftraungethinges. 

An  example  of  this  propofition  in  numbers . 

Take  any  number  as  1 3  .anddeuideitinto  two  partes  asint04.Sc  p.  Take  thefquare! 
of  13.  which  is  i6p,  take  alfo  the  fquare  of  4.  which  is  itf.andadde  thefe  two  fquare* 
together  which  make  185  .Then  multiply  the  whole  number  1 3 .  into  4.  theforefayde 
part  twife,and  you  fhall  produce  5  2.  and  5 2 :  take  alfo  the  fquare  of  the  other  part, that 
i  s,of  p .  wh  ich  is  8  r  .And  adde  it  to  the  produdes  of  r  3  .into  4.twife,that  is, vnto  5  2  .and 
52.  and  thofe  three  numbers  added  together  fhall  make  185.  whicheis  equall  to  the 
number  compofed  of the  fquares  ofthc  whole  and  of  one  of  the  partes,  which  is  alfo 
1 85  .As. you  fee  in  the  example* 


ofEuclides  Elementes.  Fol.yu 


'The  whole 

>3 

it) 

»-partes  of  the  whole* 

Multiplication  of  the 
whole  into  it  felfe. 

n 

*3 

39 

13 

1 6p 

1 6$ 

1 6 

Multiplication  of  one  of 
the  partes  into  it  felfc. 

*< 

** 

• 

4 

4 

id 

n 

4, 

185 

'The  number  compo fed  of 
the  fquares  ofthe  whole, 
and  of  one  of  the  partes 

Multiplicationof  the 
whole  into  the  forefayde 
part  twife. 

52 

53 

4 

5 * 

Si 

nequall  to 

5* 

Multiplication  of  the  o- 
ther  part  into  it  felfe* 

9 

9 

8i 

585 

_ the  number  compofedof 
the  whole  into  the  fore- 
faid  part  twife,and  ofthe 
fquare  ofthe  other  part. 

The  demonftration  ^heroffolloweth  In  Barlaam. 


The feuenth  proportion. 

tfa  number  be  deuided  into  two  numbers:  the fquare  of  the  "Whole  number  together  With  the 
fquare  of  one  of  the  partes,  is  equallto  the  fuperficiall  number  produced  of  the  multiplication  of  the 
Whole  number  into  theforefaid  part  twife, together  With  the [quart  of  the  other  parti 

Suppofe  that  the  number  tAB  be  deuided  into  the  number^  a^/Cand  CBt  Then  1 
fay  that  the  fquare  numbers  of  the  numbers  BA  and  AC  are  equall  to  the  fuperficiall 
number  produced  of  the  multiplicatio  of  the  number  B  A  into  the  number  AC  twife, 
together  with  the  fquare  of  the  number  B  C.For  forafmuch  asfby  the  4.of  this  booke)  3 

the  fquare  of  the  number  AB  is  equall  to  the  fquaresofthe  numbers  B  Cand  CA,znd 
to  the  fuperficiall  number  produced  of  the  multiplication  of  the  numbers  *3  C  and  C 
A  the  one  into  the  other  twife:  adde  the  fquare  of  the  number  C  common  to  them 

both.  Wherfore  the  fquare  ofthen  umbered  .5  together  with  thefquare  of  the  num¬ 
ber  AC  is  equall  to  two  fquares  of  the  number  A  C  and  to  one  fquare  of  the  number  C 
By  and  alfo  to  the  fuperficiall  number  produced  of  the  multiplication  of  the  numbers 
B  C  and  C  A  the  one  into  the  other  twife.  And  forafmuch  as  the  fuperficial  number  pro 
ducedofthe  multiplication  of  the  numbers  B  A  and  C  A  the  one  into  the  other  once,  5 
is  equall  to  thefuperficiall  nuber  produced  of  the  multiplication  of  B  C  into  C  A  once, 
and  to  the  fquare  of  the  number  C  A(by  the  third  of this  booke  Jbtherforc  the  number 
produced  ofthe  multiplication  of B  A  into  *AC  twife  is  equall  to  the  number  produ¬ 
ced  of  the  multiplication  ofB  Cinro  C^twife,andalfo  to  two  fquaresofthe  number  C 
A.Addc  the  fquare  number  ofSCcommon  to  them  both.Wherfore  two  fquares  ofthe 
number  C  and  one  fquare  of  the  number  C2?  together  with  the  fuperficiall  number 

T.iij.  pro-  A 


■«»  tP 


duced  of  the  multiplication  of  B  Cinto  C./?  twife  are  equall  to  the  fuperficiall  number 
produced  of  the  multiplication  of  the  number  BA  into  the  number  AC  twife  toee- 
ther  with  the  fquare  of  the  number  C  B -Wherfore  the  fquare  of  the  number  A  B  toge¬ 
ther  with  the  fquare  of  the  nubcr  AC  is  equal  to  the:  fuperficial  nuber  produced  of  the 
multiplication  ofthe  number  5  -4 into  the  number  A  C  twife, together  with  the  fquare 
of  the  number  CB.  If  therfore  a  number  be  deuidedinto  two  numbers  &c.  which  was 
required  to  be  demonftrated. 


b 


f 


6 4  the  fquare  ofthe  whole 


% 


2  5  fthe  fquare  of  the  part 

j1Isut“4 


8 

y 


^4  the  fquare  ofthe  whole  AJS 
25  the  fquare  ofthe  part  A  C 

80  the  fuperficiall  number 
9  the  fquare  ofthe  other  part  B  C 

§7 


l 


4°  4° 

80  the  fiiperficial  number  produced  of  the  multiplication  of  the 
whole  into  the  part  twife 

$ 


.  9.  r  the  fquare  of  the  other  par  t. 


7* he  Theorems.  The  ftSPropoftidn, 


If  a  right  line  he  deuided  by  chance  foe  rectangle  figure  com** 
prehended  Under  the  whole  and  one  ofthe partes foure  times, 
together  With  the fquare  which  is  made  ofthe  other parte,  is 
equall  to  the  fquare  which  is  made  ofthe  whole  and  the  for e^ 
Jaid part  as  of  one  line.  -  ■ 


Vppofe  that  there  he  a  cert  ay  ne  right  line  A  and  Jet 

it  he  deuided  hy  chaunce  in  the  point  C.  Then  1 fay  that 
the  re  hi  angle  figure  comprehended  Milder  A  B  and  BC 
foure  tymes  together  M?ith  the  fquare  Mohichismade  of 
Aids  equall  to  the  fquare  made  of A  B  and  BCas  of  one 
lin/, Extend  the  line  A  B  (hythefecondpeticion).  And 
(j  (hy  the  third  ofthe firft)  Mnto  C  B  put  an  equall  lyne  B 
And  (hy  the  46.of  the  firft)  deferihe  Mppon  AD  a 
fquare  AEF  D.And  deferihe  a  double  figure .  And  forafmuch  as  CB  is  equall 
VntoBDftttt  CB  is  eqmlhnto  GJ^  (by  the^of  the  firft)  and  like  wifeBD 


ofEuclides  Elementes . 


Fol.Ji; 


N, 

,/Y1 

M  /  G 

/A 

K 

\X  0  / 

/  V 

P  ' 

1 _  i 

ts  equallvnto  therefore  G  Ifalfo  is  equally  nto  h(K:and  by  the  fame 
reafonalfo  B  is  equall  Vnto  %0.  And  forafmuch  as  BC  is  equallvnto  B 

and  G  Kgvnto  LfKy  tberfore  (by  the  ^  6. of  the  fir  H)  the  figure  C  is  equall 
Vnto  the  figure  JfiJD}and  the 

figure  G  %ts  equall  Vnto  the  A  c  "B  i> 

figure  (RJSl,  (But  (by y  4i,ofy  ' 
u)the  figure  C  Kjs  equall  Vn 
to  the  figure  BJSL:  for  they 
are  the  fupplementes  of  the 
parallelograme  C  0.  VVher* 
fore  the figure  D  alfo  is  e* 
quail  Vnto  the  figure  ISL  Bj 
VVherefore  the/e  figures  2) 

KJCKJG  BJBJSfyare  equall 
the  one  to  the  other  .Where* 
forethofe  foure  are  quadru * 
pie  to  the  figure  C^.  Agayne 

forafmuch  as  C  B  is  equal  Vn*  ^  M  L  F 

to  BDjout  BID  is  equaUvn* 
toBKjcbatisjmtoCG.And 

C  Bis  equall  vnto  G  K^that  is  Vnto  G  B:  tberfore  C  G  is  equal!  Vnto  G  B.  And 
forafmuch  asCG  is  equalhnto  G  B^and  B  BJs  equall  Vnto  BJlytherefore  the 
figure  A  G  is  equallvnto  the  figure MB  ^and  the  figure  B  Lis  equall  Vnto  the 
figure  Bfd.But  the  figure  MB  is(by  the  4%  ofthefirU)  equalhnto  the  figure 
F  Lfor  they  are  the  fupplementes  of  the  paratlelogramme  M L:  wherfore  the 
figure  alfo  A  G  is  equall  vnto  the  figure  BJ^.  Wherfore  the fe  foure  figures  A 
GyMBfB  L,and  BJd are  equall  the  one  to  the  other:  wherfore  thofe  foure  are 
quadruple  to  the  figure  A  G.And  it  is  proued,that  thefe  foure  figures  CJfififfi) 
G  B,-)fA.'HAre  quadruple  to  the  figure  CKJ/ A  her  fore  the  eight  figures  whid > 
contayne  the  gnomon  S  T  V^are  quadruple  to  the  figure  A  l\.  And  forafmuch 
as  the  figure  A  IQ  is  that  which  is  comay ned  Vnder  the  lines  A  B  and  B  D,for 
the  line  B  KJs  equallvnto  the  line  B  D :  tberfore  that  whiche  is  contayned  Vn * 
derthe  lines  A  B  and  B  V  foure  tymes  is  quadruple  Vnto  the  figure  A  And 
it  is  proued  that  the  gnomon  S  TV  is  quadruple  to  AAf  Wherfore  that  which 
is  contayned  Vnder  the  lines  A  B  and  BD  foure  tymes  is  equallvnto  the  gnomo 
S  T  VJBut  the  figure  X  Hwhich  is  equall  to  the  fquare  made  of  AC  common 
Vnto  them  both.Wherfore  the  re  dangle figure  comprehended  Vnder  the  lines 
A  B  and  B  D  foure  tymes  together  With  the  fquare  which  is  made  of  the  line  A 
Cfis  equall  to  the gnomon  ST  Vyand  Vnto  the  figure  X  H.  But  the  gnomon  $ 
'TV:  and  the  figure  X  Hare  the  whole  fquare  ABF  D,  which  is  made  of  A 
T> :  wherfore  that  which  is  contayned  Vnder  the  lines  A  B  and  B  D  foure  times 
together  with  the  fquare  which  is  made  of  A  C,  is  equall  to  the  fquare  which  is 

Tmu,  made 


made  of  A  7). But  B  D  is  equalhmtoB  C.Wherfore  thereBangle  figure  con* 
tajnedfoure  tymes  Vnder  A  ’Band  B  C  together tyith  the  {quarts  which  is  made 
°f  A  Cjs  equall  Vnto  the  fquare  Trhich  is  made  of  A  ID, that  isyVnto  that  ftbiche 
is  made  of  A  B  andB  C  as  of  one  line .  If  therefore  a  right  lyne  be  deuided  by 
chaunce ,  the  relf angle  figure  comprehended  lender  the  Tbhole  and  one  of  the 
partes  foure  tymes  ^together  faith  the  fquare  which  is  made  on  the  other part,k 
equal!  to  the  fquare  Tabid  is  made  ofthelphole  and  the forefaid  party  as  of  one 
line>D?bicbl!>as  required  to  be  demonstrated-  ;  i 

An  example  of  this  Propofition  in  numbers. 

Take  any  number  3  s  iy.anddeuideit  into  two  partes,as  into  6.and  n.  And  multi¬ 
ply  i7.into5.namelyoneofthepartesfouretymes,andyoufhallproduce  102.  102* 
102. and  io  2.Takealfo  the  fquare  of  1 1. the  other  part,which is  121:  andaddeitynto 
the  foure  numbers  produced  of  the  whole  17. into  the  part  5.foure  tymes,&you  ihall 
make  <>  29. Then  adde  the  whole  number  iy.to  the  forefaid  part  6.  which  make  23 .  Ss, 
take  the  fquare  of  23*  which  is  5  29.  which  is  equall  to  the  number  compofed  of  the 
whole  into  the  fayd  partfoure  tymes,  aud  of  the  fquare  of  the  other  part,  which  num¬ 
ber  compofed  is  alfo  5  zp.  As  you  fee  in  the  example.  f 


-  The  whole. 


6 


102 


j  j  partes  of  the  whole 


Multiplication  of  the 
Whole  into  one  of  the 
partes  foure  times. 


X 

c.;:-L ~  ''  -  ■: 

Multiplication  of  the  o- 
ther par t  into  it  felfe. 


Addition  of  the  whole  in¬ 
to  the  part. 


Multiplication  of  the  nu- 
ber  copofed  of  the  whole 
and  the  forefaid  part  into 
it  felfe. 


17 

6 

102 

17 

6 

102 

17 

6 

102 

1  r 

I  r 

I I 
1 1 

!I2I 


17 

6 

23 

2  i 

23 
\6 9 

$29 


ioz 

102 

102 

102 

121 

5  29 


•  | 

1  :  fv' 

f .  ' 

•  ;  ■  f  ;v 

V  h  •  c 


'  | 


the  number  compofed  of  the 
whole  into  one  of  the  partes 
foure  tymes,  &  of  the  fquare 
of  the  other  part 


v  equal  to 


-  ■>  s-' 


the  fquare  of  the  number  co- 
’  pofed  of  the  whole  &  the  fore 
faidpart.  *  The 


ofEuclides  Elements* . 

The  demonftrationvheroffoilovethin  Barlaam. 
The  eight  proportion. 


Fol.  73. 


D 


If 4  number  be  derided  into  typo  numbers, the fuperficiall  number, produced  of  the  multiplication 
of t  he  whole  into  one  of  the  partes foure  tymes, together  With  the fqtiare  of  the  other  parte, is  equall  to 
the fquare  of  the  number  tempo  fed  of  the  whole  number  and  the  forefay d  part. 

Suppofe  that  the  number  AB  be  deuided  into  two  numbers  AC  and  C  B.  Then  / 
fay  that  the  fuperficiall  number  produced  of  the  multiplication  of  the  number  A  B  in 
to  the  number  C  B  foure  tymes  together  with  the  fquare  of  the  number  A  Cfs  cquall 
to  the  fquare  of  the  number  compofed  of  the  numbers  AB  &  CB.  For  vnto  the  num¬ 
ber  2?  Clet  the  number  2?  D  be  equall.  Now  forafmuch  as  the  fquare  of  the 
number  AD  is  equal  to  the  fquares  of  the  numbers  A  B  and  2>  D,Sc  to  the 
fuperficiall  number  produced  of  the  multiplication  of  the  numbers  AB  Sc  2 
B  D  the  one  into  the  other  twifef  by  the  q-.of  this  booke)  And  the  numb  er 
2?Z>is  equall  to  the  number B  C\  therefore  the :  fquare  of  the  number  AD  2 
is  equall  to  the  fquares  of  the  numbers  A  B  and  B  C,  and  to  the  fuperficiall 
number  produced  of  the  multiplication  of  the  numbers  AB  and  SC  the 
one  into  the  other  twife.But  the  fquares  of  the  numbers  A  B  and  B  C  are  e- 
quall  vnto  the  fuperficiall  number  produced  of  the  multiplication  of  the  6 
numbers  A  B  and  B  C  the  one  into  the  other  twife,  and  to  the  fquare  of  A 
C(by  the  former  propofition)  Wherfore  the  fquare  ofthe  number  ADis 
cquall  to  the  fuperficial  number  produced  of  the  multiplication  of  the  nu- 
bers  A  S  and  B  C  the  one  into  the  other  foure  tymes, and  to  the  fquare  of  [ 

the  number  A  C.But  the  fquare  of  the  number  A  D  is  the  fquare  ofthe 
number  compofed  ofthe  numbers  .^2?  and  2?  C:  for  the  number#  D  is  e- 
qual  to  the  number  B  C. Wherfore  the  fquare  of  the  number  compofed  of  thenumbefs 
AB  andS  C  is  equall  to  the  fuperficiall  number  produced  ofthe  multiplication  of  the 
numbers  A  B  and  B  Cthe  one  into  the  other  foure  tymes,  &  to  the  fquare  of  the  num¬ 
ber  A  C./f therfore  a  number  be  dcuidedinto  two  numbers, &c- 


8 

2 


1 6 
v___ 


8 

2 

1 6 


8 

2 


1 6 


~y — 


8 

2 

1 6 


6 

6 


64.  the  fuperficiall  number  produced  of  the  multipli-  _N 

cation  ofthe  numbers  AB  and  B  Cthe  oneinto 
the  other  foure  tymes. 


4  3  6  the  fquare  of  A  C, 


10 

10 

too 


2 1 

100 


> 


thejquare  of  the  number 
compofed  o  iAB  and  B  C. 

the  fuperficial  nuber produced  ofthe  multiplicatiS  made  4*times 
the  fquare  number  of  A  C, 


ConHmlHon. 


DemonHru-* 

non. 


*Ihe  fecondUfwhp 

Thep.Theoreme.  ThepfPropofitm. 

'  '  '  -  1  1  '  '  ■  •  '-'ey  >\  I  j'< 

Sfe?  If  i  right  line  be  dcuided into  two  e  quail  partes,  and  into 
Wo  lone  quail  partes,  the fquares  which  are  made  of  the*vne~ 
quail  partes  of  the  whole, are  double  to  the  fquares, which  are 
made  of  the  halfe  lyne^and  ofthatlyne  which  is  betwene  the 
feUions . 

Vppofe  that  a  certayne  right  line  A  3  he  dtuided  into  two  equaS 
partes  in  the  pointe  C,  and  into  t"Wo  Vnequall partes  m  the  pointe  <D. 
Then  I/ay  t  bat  the Jquares  "which  ar  e  made  of  the  lines  A  F)  and  S) 
3  .are  double  to  the/quarts  whiche  are  made  of  the  lynes  A  C  and  C 
JD.For(by  tfe  1 \,  of  the 
fir  ft  )er  eel  from  y  point 
C  to  the  right  UneABa 
perpendiculer  line  C  E. 

And  let  C  E(by  the  $,of 
the  fir  ft)  be  put  e  quail 
Vnto  either  ofthefe  lines 
ACzsrCB:  and(by  the 
fir  ft  peticio)  draw  lines 
from  A  to  E,  and  from 
E  toB.And (by  the  5  1 . 
of  the  fir  ft)  by  the  point 
l D  dra"W  Vnto  the  line  EC  a  parallel  lyne,and  let  the fame  be  IDF:  and  (by  the 
felfe  fame)  by  the  point  E  dra"W  Vnto  A3  a  line  paralleled  let  the fame  be  FG , 
And  ( by  the  first  petkion)draw  a  line  from  A  to  F,  And  forafmuch  as  AC  is  e* 
quail  Vnto  C  Ejberforefby  the  5  of  the  fir  ft)  the  angle  EACis  equal  Vnto  the 
angle  C  E  A  And  forafmuch  as  the  angle  at  the  point  C  is  a  right  angle:  therfore 
t/c  angles  remay  n’wgE  A  f  and  A  E  C3aree quail  vnto  one  right  angle,  "Where* 
fore  eche  ofthefe  angles  EAC  and  AECis  the  halfe  of a  righ  t  angle.  And  by 
the  fame  reafon  alfoecbe  ofthefe  angles  EfB  Cand  C  E  Bis  the  halfe  of  a  right 
angle. VV her fore  the  "whole  angle  ABB  is  a  right  angle, And  forafmuch  as  the 
angle  G  E  F  is  the  halfe  of  a  right  angle  3but  EG  F  is  a  right  angle.  For  (by  the 
2  9  of  the  fir ft )it  is  equall  Vnto  the  in"Ward  and  oppofite  angle, that  is/Vnto  EC 
3:  "wherfpre  the  angle  remay  ning  E  F  Gis  the  halfe  of  a  right  angle  ,VVhere * 
fore( by  the  6 .1 common  fentence )tht  angle  G  EFis  equal l  Vnto  the  angle  EFG, 
yVherforeal/o  (by  the  6.0ft  he  f it j fifth  efid  e  EG  is  equdllvnto  the  fide  F  G,A# 
gains  forafmuch  as  the  angle  at  the  point  Bis  the  halfe  of a  right  angle  fut  the 
angle  F  D  B  is  a  right  angle  for  it  alfo(by  the  29  .of the firfi )  is  e  quail  V  nto  the 

irtWard 


ofEuclides  Element# s . 


inwarde  and  oppofte  angle  E  C  E.  Wherefore  the  angle  remayning  EFT)  is 
the  halfe  of  a  right  angle .  Wl:erfore  the  angle  at  the  point  E  is  equall  lento  the 
angle  E  EE.  Wherfore  ( hy  the  #>.  ofthefrH)  the  fide  E)  F  is  eqnall  lento  the 
fide  E)  E.  Andforafmuch  as  AC  is  equal l  lento  C  E>  therfore  the fiquare  "which 
is  made  of  A  C  is  equall  lento  the  fiquare " which  is  made  ofC  E.  Wherefore  the 
fquares  "Which  are  made  ofC  A  and  C  E  are  double  to  the fiquare  "which  is  made 
of  A  C.  Eut  (by  the  47*  of the  firft)  the jquare  "Which  is  made  of  E  A  is  equallto 
the fquares ' which  are  made  of  AC  and  C  E  ( For  the  angle  ACEis  a  right  aw 
gle)  "Wherefore  the fiquare  of  A  E  is  double  to  the fiquare  of A  C.  Agayne  forafi 
much  as  E  Gfis  equalllmto  G  F,  the fiquare  therfore  "Which  is  made  ofE  G  is  e* 
qual  to  the fiquare  "Which  is  made  ofiGF.  Wherfore  the  fquares  "Which  are  made 
ofGE  and  G  F  are  double  to  the  fiquare  "Which  is  made  of G  F.  Eut  ( by  the  47* 
oj  the  firU)  the fiquare  "Which  is  made  ofiEF  is  equallto  the  fquares  "Which  are 
made  of  EG  and  G  F.  Wherfore  the  fiquare  "Which  is  made  ofE  F  is  double  to 
the fiquare  "which  is  made  ofiGF.  Eut  G  F  is  e  quail  lonto  C  ID.  Wherefore  the 
fiquare  "Which  is  made  of 
EF  is  double  to  the 
fiquare " which  is  made  of 
C  E) .  And  the  fiquare 
"Whiche  is  made  of  A  E 
is  double  to  the  fiquare 
"Which  is  made  of  A  C. 

Wherefore  the  fquares 
' which  are  made  of  A  E 
and  E  F  are  double  toy 
fquares  which  are  made 
of  AC  and  CE).  Eut  (by 
the  47*  of the  firH)the fiquare  which  is  made  of  AFis  equal  to  the  fquares  which 
are  made  of  A  E  and  EF(  Forj  angle  A  E  F  is  a  right  angle ) .  Wherfore  the 
fiquare  "Which  is  made  of AF  is  double  to  the  fquares  "Which  are  made  of AC  iyC 
Ed.Eut  ( by  the  47*  o  f  the  firfl)y  fquares  "Which  are  made  ofiAE)  and  E)  F  are  e* 
quail  toy  Jquare  "Which  is  made  of A  F.Fory  angle  oty  point  E>  is  a  right  angle . 
WJ:>erfiore  the fquares " which  are  made  ofiAE)  and  E)  F  are  double  toy fquares 
"Which  are  made  of  AC  and  C  E).  Eut  E)  F  is  equall  Ipnto  E)  E.  Wherfore  the 
fquares  "which  are  made  of  A  E)  and  E)Ej  are  double  to  the  fquares  "Which  are 
made  of  AC  and  C  E).  If  therfore  a  right  line  be  deuided  into  two  equall  partes 
and  into  two  lane  quail  partes  jbe fquares  "Which  are  made  of the  lane  quail partes 
of the  "whole yare  double  to  the  fquares  "which  are  made  of the  halfe  lyne3  and  of 
that  lyne  "Which  is  betwene  the  fie  (lions:  * which  "Was  required  to  he  proued. 


E 


f  An  example  of  this propofition  in  numbers. 

Take  any  enen  number  as  1 2.  And  deuide  it  hrft  equally  as  into  <?. and  6t  &then  m» 
equally  as  into  8  ,&  4.  And  take  the  difference  of  the  halfe  to  one  of  the  vnequal  partes 

V.ij»  which 


The fecondHooke 

which  is  2  .And  take  the  fquare  numbers  of  the  vnequalJ  partes  8,  and  4,which  are 
and  itf.’and.adde  themtogether,which  make  80, Then  take  the  fquares  ofthehalfe  6, 
and  of  the  differece  2:  which  are  3  d^anciq:  which  added  together  make  40.Vn.to  which 
numberthe  number  compofed  of  the  fquares  of  the  vnequall  partes,whiche  is  So,  i# 
double. As  you  fee  in  the  example* 


The  whole. 


Multiplication  of eche 
vnequalpart  into  himfelf. 


>< 


Multiplication  of  the  half 
and  of  the  difference  eche 
into  himfelfe. 


12 


8 

8 


4 

'4 


6 

6 


ii 

2 

2 


^  J*  the  equal! pat tes 


8 


4  J 
2 


the  vnequall  partes 


64 


16 


64 

16 


80 


3  6 
4 


40 


the  difference  ofthc  halfe  to 
one  of  the  partes 


r  the  number  compofedof  the  C 
fquares  of  the  vnequal  partes  ' 


>  double  to 


the  number  compofed  of  the 
fquares  of  the  halfe,and  of  the 


difference. 

The  demonftration  wherof followeth  in  Barlaam*. 


J 


The  ninth  Tropo/ition. 

If  a  number  be  deuided  into  tVoo  e  quail  numbers, and againe  be  deuided  into  two  inequall  partes:  the 
fquare  numbers  of  the  vnequall  numbers, are  double  to  the  fquare  which  is  made  of  the  multiplicand 
on  of the  halfe  number  into  it  felfe,  together  With  the  fquare  Whiche  is  made  of  the  number  ft  bed 
tWene  them. 

For  let  the  number  zA  B  being  an  cuen  number  be  deuided  into  two  cquall  numbers 
A  C  &  C  B:  Sc  into  two  vnequall  nubers  AD  and  D  A.Then  /  fay  that  the  fq  uare  num¬ 
bers  o (AD  and  D  B,txt  double  to  the  fquares  which  are  made  of  the  multiplication 
of  the  numbers  AC  and  CD  into  themfelues.  For  forafmuch  as  the  number  si  B  is  an 
euen  number, and  is  deuided  alfo  into  two  equal  numbers  A  C and  Ci^and  afterward 
into  two  vnequal  nubers  A  D  and  DB-.t  herefore  the  fuperficial  nuber  produced  of  the 
multiplicand  of  the  nubers  AD  &DB>  thone  into  the  other, together  with  the  fquare 
of  the  number  D  C,is  equal  to  the  fquare  of  the  number  ACfby  the  fife  proportion  ) 
Wherfore  the  fuperficiall  number  produced  of  the  multiplication  of  the  numbers  AD 
and  D  B  the  one  into  the  other  twife,  together  with  two  fquaresof  the  number  CD}is 
double  to  the  fquare  of  the  number  zA  C,  Forafmuch  as  alfo  the  number  A  B  is  deui¬ 
ded  into  two  equal  numbers  zA  Cand  C3,therfore  the  fquare  number  of  AB  is  qua¬ 
druple  to  the  fquare  number  produced  of  the  multiplication  of  the  number  <sA  C  into 
it  felfef  by  the  q.propofition)  .MOfeouer  forafmuch  as  the  fuperficiall  number produ¬ 
ced  of  the  multiplication  of  thenumbers  A  the  one  into  the  other  twife  to¬ 

gether,  with  two  fquares  of  the  number  D  C  is  double  to  the  fquare  number  of  CA.&c 

forafmuch 


of  Euclides  Element?  s . 


Fol.j 5< 


forafmuch  as  there  are  two  numbers*  of  whiche  the  one  is  quadruple  to 
one  and  the  felfe  fame  number,and  the  other  is  double  to  the  fame  hum 
ber:  therefore  that  number  whiche  is  quadruple  fhall  be  double  to  that 
number  whiche  is  double-  Wherefore  the  fquare  of  the  number  A  B  is 
double  to  the  number  produced  ofthe  multiplicand  of  the  numbers  ~ 
jD  and  D  B  the  oneintotheothertwife  together  with  the  two  fquares  of 
the  number!)  C.Wherfore  the  number  which  is  produced  of  the  mul¬ 
tiplication  ofthe  numbers  A  D  and  D  B  the  one  into  the  other  twife*  is 
lelfe  the  halfe  of  the  fquare  ofthe  number  A  B  by  the  two  fquares  ofthe  ^ 
numbers  D  C.  And  forafmuch  as  the  nuber  produced  of  the  multiplica¬ 
tion  ofthe  nuber  s  AD  8c  D  B  the  one  into  the  other  tw,ife, together  with 
the  nuber  copofed  of  the  fquares  ofthe  numbers  A  D  an&T)  B  is(by  the 
4.propofiaori)equall  to  the  fquare  ofthe  number.^  A  ;  therforethe  nu¬ 
ber  compofed  of  the  fquares  of  the  numbers  A  D  8c  D  "Bis  greater  then 
the  halfe  ofthe  fquare  nuber  ofc^  B,  by  the  two  fquares  of  the  number 
I)  C.  And  the  fquare  ofthe  numbered/  2?  is  quadruple  to  the  fquare  of  6 
the  number  A  C.Wherfore  the  number  compofed  of  the  fquares  of  the 
numbers  A  D  and®  B  is  greater  then  the  double  of  the  fquare  of  the 
number  ACby  two  fquares  of  the  number  D  C.Wherfore  the  faid  num¬ 
ber  is  double  to  the  fquares  ofthe  numbers  A C and  C‘Z>.  7f therefore  a 
number  be  deuided  &c,which  was  requiredto  be  demonftrated. 


,'D 


8 

8 


2 

2 


<5q.  the  fquare  of the  vnequall  part  AD  4  the  fquare  of  the  vncquall  part  £  JD 


5 

5 


2  5  the  fquare  of  the  halfe  A  C* 


9  the  fquare  of  C D3nameiy,of  the  number  fet  betwene. 


9 


3  4  the  fquares  of  the  halfc*and  of  the  number  fet  bctwenc. 


*4 

4 


68 

34 


I2* 


~  68  the  fquares  of  the  vncquall  partes. 

.  •  ■ t-  -  -  .  *.  ■  -  ^  1  _  _  ■  _  *  * 1 «  .* '  *  •  -  -4 

The  10. Theorems.  The  xo.Trofofition . 

If  a  right  line  be  deuided  into  two  equal partes vnto  it  be 
added  an  other  right  line  direBlyithe  fquare  which  is  made  of 
the  whole  &  that  which  is  added  as  of  one  line ,  together  with 


The fecond'Booke 

the fquare  whiche  is  made  of  the  lyne  whiche  is  added \  thefe 
mo fcjuares  ( 1 fay')  are  double  to  thefe fquarcsynamelyjo  the 
Jquare  which  is  made  of  the  halfe  line to  the  fquare  which 
is  made  of  the  other  halfe  lyne  and  that  whiche  is  added  >  as 
of  one  lyne . 


ConJtrntlim. 


Dtmtnflra- 
tion . 


V ppofe  that  a  cert  ay  ne  right  line  ABbe  deuided  into  tleo  equal l  partes 
\in  the  point  C.And  Vnto  it  let  there  he  added  an  other  right  line  dir  eft* 
^^lyjiamel0d  D.  The  I  fayjhat  the  fquar.es which  are  made  of  the  lines 
*A  D  and  D  B  are  double  to  the  fquares  Which  are  made  of  the  lines  _A  Cand  C 
D.  (Rayfeyp  (by  the  n,  of  the  firH )  from  the  point  C^nto  the  right  line  ACT) 
a  perpendiculer  lyneyand  let  the  fame  be  C  E.  And  let  C  E  (by  the  5,  of  the  fir  ft) 
be  made  equallvnto  either  of  thefe  lines  J  Cand  CB.  And  (by  the  firft  petiti * 
on)  draw  right  lines  from  E  to  A^andfrom  E  to  B.  And(by  they,  of the  fir  ft) 
by  the  point  E fir  aw  a  line  parallel  Vnto  C  Dymd  let  the fame  be  E  EAnd(byy 
felfjame  )by  the  point  T)  draft;  a  line  parallel  Vnto  C  E  and  let  the fame  be  IDE. 
Andforafmuch  as  Vpon  thefe  parallel  lines  CEzs'DE  ligbtetb  a  certain  right 
line  E  Fyherf m(  by  the  zyjfthe  firft)  the  angles  CEFandEFD  are  equal 
Vnto  tft>o  right  angles jETher fore  th e  angles  FE  B^and EFD  are  lefle  then 
tit  0  right  angle  sJBut  lines  produced  from  angles  leffe  then  two  right  angles(  by 
the  fifth  peticion)at  the  length  meete  together. VTberfore  the  lines  EB  andF 
D  beyng  produced  on  that  fide  that  the  line  B  Distill  at  the  length  meete  to * 
gether: Produce  them  and  let  them  meete  together  in  the  point  G .  And  (by  the 
firft  peticion)draW  a  line  from  A  to  G,  Andforafmuch  as  the  line  A  C  is  equal l 
Vnto  the  line  C  Ey  the  angle 

alfoAECis  (by  the of  the  E  F 

firfl)equall  ’Vnto  the  angle  E 
A  Ct  And  the  angle  at  y  point 
Cis  a  right  angle.  W her fore 
eche  of  thefe  angles  E  A  C<& 
and  A  EC  is  the  halfe  of  a 
right  angle.  And  by  the  fame 
reafon  eche  of  thefe  angles  C 
E  B,  and  ETC  is  the  halfe 
of  a  right  an fie.  Wherefore 

the  angle  A  E  Bis  a  right  angle.  And fotafmuch  as  the  angle  ETC  is  the  halfe 
of  a  right  angle  fherfore  (by  the  1$.  of the  firH)tbe  angle  DBG  is  the  half  of  a 
right  angle.  But y  angle  BDGis  a  right  angle(for  it  is  equal  Vnto  the  angle  D 
CEfor  they  are  alternate  ahgles)Wher fore  the  angle  remaining  DG  B  is  the 
halfe  of  a  right  angle. VV her  fore  (by  the  6  common  fentence  of  thefirfl)the  an * 
gleDGB  is  equall  to  the  angle  DBG #  VF her  fore  (by  the  6  .of the firft)  the 

fide 


m  is  equall  Imto  the  fide  GD.  Agayne  forafmuch as  the  angle  E  G  F  it  thi 
balfeofci  right  angle :  and  the  angle  at  thepointe  F is  a  right  angle:  for  (by  the 
3Fof the  first)  it  is  equall  lento  the  oppofte  angle  EC  ID.  Wherefore  the  angle 
remay  ning  F  EG  is  the  halfe  of a  right  angle.  Whet  fore  the  angle  E  G  F  is  e* 
quail  to  the  angle  F  E  G.  Wherfore  ( by  the  6.  of the  frit  the  fide  FF  Eis  equall 
lento  the  fide  F  G.  And  forafmuch  as  EC  is  equall  Imto  C  A,  the  fquare  dfe 
i! Ariel:  is  made  ofE  C  is  equall  to  the  fquare  which  is  made  of  C  A*  Wherefore 
the fquares  which  are  made  of  CEand  C  A  are  double  to  the  fquare  "Which  is 
made  of  AC.  Eut  the fquare  "Which  is  made  ofE  A  is  (by  the  47*  of  the  fir  A)  e* 
quail  Imto  the fquares  which  are  made  of  EC  and  C  A.  Wherefore  the  fquare 
Which  is  made  of  E  A  is  double  to  the  fquare  which  is  made  of  A  C.  Again  e  for* 
afmuch  as  G  F  is  equall  Imto  E  F/he  fquare  alfo  which  is  made  ofGF  is  equall 
to  the  fquare  winch  is  made  ofFE.  VVherfore  the fquares  which  are  made  of  G 
F  andE  F  are  double  to  the fquare  which  is  made  ofE  F.  Eut  (by  the  47  •of  the 
fir  A)  the fquare  which  is  made  of EG  is  equall  to  the  fquares  which  are  made 
of GF andE  F.  Wherefore 
the  fquare  which  is  made  of 
EG  is  double  to  the  fquare 
Which  is  made  ofE  F.EutE 
Fis  equall  Imto  C  D,  wher- 
forey  fquare  which  is  made 
ofE  G  is  double  to  the fquare 
Which  is  made  of  C  D.  And 
it  is  proued  the  fquare  which 
is  made  of  E  A  is  double  to 
the  fquare  which  is  made  of 
A  C.  Wherfore  thefquar  es  Which  are  made  of  A  E  andE  G  are  double  to  the 
fquares  which  are  made  of  A  C  and  C  D.  Eut(by  the  47-  of ths fir  A)  the fquare 
which  is  made  of  AG  is  equall  to  the  fquares  which  are  made  of  A  E  and  E  G, 
Wherefore  the  fquare  which  is  made  of  AG  is  double  to  the fquares  which  are 
made  of  AC  and  C  ID.  Eut  Imto  the  fquare  which  e  is  made  of  AG  are  equall 
the  fquares  which  are  made  of A  D  and  D  G. Wherfore  the  fquares  which  are 
made  of  A  D  and  D  G  are  double  to  the  fquares  which  are  made  of  A  C  andD 
C  •  Eut  D  G  is  equall  Imto  D  E.Wherfore  the fquares  which  are  made  of  A  D 
And  D  E  are  double  to  the fquares  which  are  made  of  AC  and  D  C.  Iftherfore 
aright  line  be  deuidedinto  two  equall partes y  and  Imto  it  be  added  an  other dyne 
directly ,  the  fquare  which  is  made  of  the  Whole  and  that  which  is  added ,  as  of 
one  line  together  With  the  fquare  which  is  made  of  the  line  which  is  added ghefe 
two  fquares  (I fay)  are  double  to  thefe fquares  y  namely }  to  the fquare  which  is 
made  of  the  halfe lyne,  and  to  the  fquare  which  is  made  of  the  other  halfe  lyne 
and  that  which  is  added gs  of one  lyne:  which  Was  required  to  be  proued. 

v  •  .  ...  •  -  •- .  .  •'  . 

f  An  other  demonAration  after  Eelitarius. 

V.iili. 


Suppofe 


The  fecondBooke 

Suppofe  that  the  lyne  AB  be  deuided  into  two  equal!  partes  in  the  poytiteC. 
And  vnto  it  let  there  be  added  an  other  right  lyne  dire&ly,  namely,  B  D.  Then  I  fay 
that  the  fquare  of  AD  together  with  the  fquare  of  B  D  is  double  to  the  fquares  of  A 

Vpon  the  whole  line  AD  defcribe  a  fquare  AD  E  F.And  ypon  the  halfe  Ivne  A  Cde-? 
fcnbe  the  fquare  A  C  G  H.And  produce  the  Tides  G  H  and  C  H  till  they  cut' the  lides  E 
F  &  D  F,wherby  ilialbc  defcribed  the  figure  H  L  K  F, which  ihalbethe  fquare  of  the  line 
C  D  the  Corollary  of  the  4.  of  this  boke,&  by  the  34.Propofition  of  the  i.)itis 

manifcfhf  we  draw  the  diameter  C  D.  For  the  lyne  K  F  is  equall  to  the  line  C  D.  And 
making  alfo  the  lines  H  M  and  H  N  equall  to  either  of  thefe  lynes  A  C  and  C  B,  drawe 
the  lynes  M  G  and  NP  cutting  the  one  the  other  right  angled  wife  in  the  point' Q.  Ei¬ 
ther  of which  lynes  let  cutthefides  of  the  fquare  A  D; 

E Fin  the  pointes  OandP.Nowitnedeth  nottoproue 
that  the  figure  H  Qis  the  iquare  of  the  lyne  A  C,fcyng 
that  it  is  the  fquare  of  the  line  C  B  :  as  the  figure  QJ  is 
the  fquare  ofthe  line  BD  ;  neither  alfo  needethit  to 
proue  that  the  paralleiograme  H  P  is  eq  uall  to  either  of 
the fupplementes E  H  and H D  :  nor  that  the  fupplc- 
mentcs  N  O  and  QJ.  are  equall.  For  all  this  is  manifeft 
cue  by  the  forme  of  the  figure, for  that  all  theangles  a- 
bou  t  the  diameter  arc  half  right  angles,  &  the  fidcs  are 
equall.  Wherfore  if  we  diligently  marke  of  what  partes 
the  fquare  HF  which  is  the  fquare  of  CD,  iscompo- 
fed,we  may  thus  reafo.  Forafmuch  as  the  whole  fquare 
E  D  is  compofed  of  the  two  fquares  A  H  and  H  F  and  of 
the  two  fupplementes  E  H  and  H  P,we  muftproue  that 
thefe  fupplementes  with  the  fquare  QJJwhich  is  the  fquare  of  the  line  B  D)  are  equall 
to  the  two  fquares  A H  and  H  F.For  then  (hall  we  proue  that  thefe  two  fquares  AH  6£ 
HF  taken  twile  are  equall  to  the  whole  fquare  DE  together  with  the  fquare  of  QF9 
which  thing  we  tookefirftin  hand  to  proue. And  thus  do  I  proue  it. 

The  Supplement  E  H  is  equall  to  the  paralleiograme  H  P.  And  the  fquare  A  H  to¬ 
gether  with  the  lefier  fupplemet,  N  0,is  equall  to  the  other  fupplemet  H  D  (by  the  firft 
common  fentence  fo  oftentymes  repeted  as  is  neede )  wherfore  the  two  fupplementes 
E  H  and H  D  are  equall  to  the  fquare  A  H  and  to  the  Gnomon  K  H  L  P  QjO.  If  therfore 
vnto  either  of  them  be  added  the  fquare  QJ :  the  two  fupplementes  E  H  and  H  D  to¬ 
gether  with  the  fquare  of  QJ  dial  be  equal  to  the  fquare  A  H,  &  to  the  Gnomon  K  H  L 
P  QO  and  to  the  fquare  QJJBut  thefe  three  figures  do  make  the  two  fquares  A  H  and 
H  F.VVherforc  the  two  fupplementes  E  H  and  H  D  together  with  the  fquare  QJF  arc  e- 
quall  to  the  two  fquares  AH  and  H  F, which  was  the  fecond  thing  to  be  proued. Wher¬ 
fore  the  two  fquares  A  H  and  H  F  beyng  taken  twife  are  equall  to  the  whole  fquare  D 
E  together  with  the  fquare  of  QJ3.  Wherfore  the  fquare  D  E  together  with  the  fquare 
QJ  is  double  to  the  fquares  A  H  and  H  F:  which  was  required  to  be  proued. 


f  jin  example  of  this  Tropofition  in  numbers. 

Take  any  euen  number  as  1 8:  and  take  the  halfe  of  it  which  is  9.  and  vnto  18.  the 
whole, adde  any  other  number  as  3  .which  maketh  2 1 .  Take  the  fquare  number  of  1 1 . 
(the  whole  number  and  the  number  added)  which  maketh  441.  Take  alfo  the  fquare 
of  3  C  the  number  added)  which  is  p.  which  two  fquares  added  together  make  450. 
Then  adde  the  halfe  number  p.  to  the  number  added  3.  which  maketh  12.  And  take 
the  fquare  ofp.the  halfe  number  and  of  1 2,  the  halfe  number  and  the  number  added 
which  fquares  are  8 1.  and  144.  and  which  two  iquares  alfo  added  together  make  235; 
■vnto  which  fumme  the  forefayd  number  450,  is  double.  As  you  fee  in  the  example. 


The 


The  whole. 

The  number  added. 


ententes • 

r  18 


Fol.yj. 


Multiplication  of  the  whole  and 
the  number  addedinco  himfelf. 


Multiplication  of  the  number 
addeddnto  himfelfe. 


Multiplication  of  the  halfe  in¬ 
to  himfelfe. 


Multiplication  of  the  halfe,  & 
the  number  added  into  it  felfe. 


>■«< 


l 


21 

21 

21 

21 


42 


44I 

44I 

3  9 

.  - 


9 

9 


81 

12 
1 2 

24 

12 

144 


1 


the  halfe- 


81 

iH 

225 


~  The  number  compofed  of 
the  fquare  of  the  whole  & 
the  number  added  and  of 
the  fquare  of  the  number 
added, 

-doublet© 


the  number  compofed  of 
the  fquare  of  the  halfe  and 
of  the  fquare  of  the  halfe 
and  the  number  added. 


The  demonftration  wheroffolloweth  in  Barlaam* 
r.  -  The  tenth  (Proportion, 

If  ten  tnmnomber  be  deuided  into  two  squall  nombers ,  and  vnto  it  be  added  any  other  nomber:  the 
fquare  nomber  of  t  he  Whole  nomber  compofed  of  the  nober  and  of  that  which  is  added ,  and  the 
fquare  nomber  of  the  nober  added.-thefe  tWo  fquare  nobers(  I  fay  )added  together ,  are  double  t@ 
thefe fquare  nombers, namely, to  the  fquare  of  the  halfe  nomber t  and  to  the  fquare  of  the  nomber 
compofed  of  the  halfe  nomber  and  of  the  nomber  added. 

Suppofe.  that  the  nomber  c A  B  being  an  euen  nomber  be  deuided  into  two  cquall 
nombers  AC  andCZ? :  and  vnto  it  let  be  added  an  other  nomber  BD  .  Then  I  fay,  that 
the  fquare  nombers  of  the  nombers  AD  an  dDB  are  double  to  the  fquare  nombers  of 
*XC  and  CD.  For  forafmuch  as  the  nomber  AD  is  deuided  into  the  nombers  AS  and 
BT):  therefore  the  fquare  nombers  of  the  nombers  AD  and  DB  are  equall  to  the  fu~ 
perficiail  nomber  produced  of  the  multiplication  of  the  nombers  AD  andD2?  the  on 
into  the  other  twife,together  with  the  fquare  ofthe  nomber  .^.5 f  by  the  7  propofitio) 
But  the  fquare  of  the  nomber^  is  equal  tofowerfquaresof  either  ofthe  nombers 
ACotCB  (for  ^4Cis  equall  to  the  nomber  CB ):  wherforealfo  the  fquares  ofthe  nom¬ 
bers  AD  an  dDB  are  equall  to  the  fuperficiall  nomber  produced  ofthe  multiplication 
of  the  nombers  AD  and^JDi?  the  one  into  the  other  twife,  and  to  fower  fquares  ofthe 
nomber  BCox  CA,  And  forafmuch  as  the  fuperficiall  nomber  produced  ofthe  multi¬ 
plication  of  the  nombers  AD  and  DB  the  one  into  the  other, together  with  the  fquare 
ofthe  nomber  Ci?, isequal  to  {quareofthe  nomber  CD(by  the  6  propofitio) :  therfore 
the  nomber  produced  of  the  multiplication  of  the  nombers  AD  and  DB  the  one  into 
the  other  twife  together  with  two  fquares  of  the  nomber  CB,is  equall  to  two  fquares 
ofthe  nomber  CD .  Wherefore  the  fquares  of  the  nombers  AD  and  DB  are  equall  to 

X-£»  two 


rJL 


The fecondTHookf 

two  fquares  ofthe  nomber  C  ® ,  and  to  two  fquares  of  the  nomber  AC .  Where>- 
fore  they  are  double  to  the  fquares  ofthe  numbers  AC  and  C®.  And  the  fquare 
of  the  nomber  <s A  D  is  the  fquare  of  the  whole  and  of  the  nomber  added  ;  And 
the  fquare  of  ®  B  is  the  fquare  of  themombe  r  added :  the  fquare  alfo  of  the  nomber  CD 
is  the  fquare  of  the  nomber  compofed  of  the  halfe  and  of  the  nomber  added  :  If  there¬ 
fore  an  euen  nomber  be  deuided.&c.  Which  was  required  to  be  proued. 


8 

8 


2 

2 


the  fquare  of  A  D 


4  the  fquare  of®  B 


the  fquare  of  C  ®,namely.,of  the  number  compofed  of  the  halfe  and 
of  the  number  added. 

3  ay  Vd-f"1 


J 

9 


the  fquare  ofthe  halfe  A  C. 


2$ 

9 

34 


k 


68 


J 


■  rs8| 
J  341 


2. 


A. 


yk^Tbe  I,  Trobleme%  TheufPropoftion . 

To  deuide  a  right  line  geuen  in  fuch  fort,  that  the  reBangU 
fgure  comprehended  ynder  the  whole >and  one  ofthe  partes , 
/hall  he  e  quail  vnto  the  fquare  made  ofthe  other part . 

fppoje  that  the  right  line  geuen  he  A  <B.  "Flow  it  is  required  to  deuide 
||  the  line  A  <B  in  fuch forty  that  the  re  Bangle  figure  contajned  Tmder 
the  vhole  and  one  of the  partes  yfball  be  equal l  Ivnto  the  fquare  “Which 
is  made  of the  other  part.lDefcribe  (by  the  46.  of the  fir ji)  'Vpon  A  35 
Conttrutlion.  a  fquare  ABCftD.  And  (  by  the  10.  of  the 
firjl  )  deuide  the  line  A  C  into  two  equall 
partes  in  the  point  E,and  draw  a  line  from 
B  to  E.  And  (by  the  fecond petition)extend 
C  A  lento  the  point  F  .And  (by  the  3.  of the 
firfi)put  the  line  E  F equall Tmtoj  line  B  E. 

And  (by  the  4 6.  of  the firft)vpon  the  line  A 
F  defenbe  a fquare  FG  AH.  And  (by  the 
2.  petition)  extend  GH  Imto  the  point  Ff 
T hen  I  fay  that  the  line  yfB  is  deuided  in 
the point  H  in  fuch  fort,  that  the  reRangle 
figure  -which  is  compreheded  Tender  A  B  and 
B  FI  is  equall  to  the  fquare  -which  is  made  of 
Demon  ft  ratio  AH.  For  forafmuch  as  the  right  line  A  C 
is  deuided  into  two  equall partes  in  the poynt 
E,and  'Vnto  it  is  added  an  other  right  line 

jK 


G  _p 

% 

H  A 

E 

V 

K  C 

of Euclides  Element es .  Fol.jS . 

^F.  Therefore  (by  the  6. of  the fecond)  the  re&angle  figure  contayned  Wider 
C  F  and  F  A  together  loith  the fijuare  1 vhich  is  made  of  A  E  is  equall  toy  fquare 
lohichis  made  ofR  F .  But  E  F  is  equall  Wito  E  B .  VVherefore  the  re  Bangle 
figure  contayned lender  C  Fynd  F  A  together  loith  the  fquare  lohich  is  made  of 
E  A  is  equall  to  the  fquare  lohich  is  made  of  EE.  'But  (  by  47.  of  the  firfi  ) 

Wito  the  fquare  "Which  is  made  of  EB  are  equall  the  fquares  lohich  are  made  of 
B  AandAE .  For  the  angle  at  the  poynt  A  is  a  right  angle  .  VVherefore  that 
lohich  is  contayned  "wider  C  F and  F  A}  together  loith  the  fquare  lohich  is  made 
of  A  E,  is  equall  to  thefquares  lohich  are  made  of  BA  and  yfE .  T ake  away 
the fquare  lohich  is  made  of  A  E  lohich  is  common ,  to  them  both :  VVherfore 
the  reElangle  figure  remayning  contayned  Wider  CF  and  F  A  is  equall  Wito 
the  fquare  lohich  is  made  ofAB .  And  that  lohich  is  contained  Wider  the  lines 
C  F  and  F  A  is  the  figure  F  If .  For  the  line  F  A  is  equall  Unto  the  line  F  G. 

And  the fquare  lohich  is  made  of  A B  is  the  figure  A  ID .,  VVherefore  thefi * 
gure  F  if  is  e quail  Wit  0  thefigure  AD  .  Takeaway  the  figure  A  if  lohich 
is  common  to  them  both.  VVherefore  the  reftdue>  namely }  thefigure  FH  is 
equall  Unto  the  rejidue  ynamely}Unto  thefigure  Elf)  .  But  the  figure  HAD  is 
that, lohich  is  contayned  Wider  the  lines  AB  and  BH,for  AB  is  equation* 
to  BD .  And  thefigure  F  His  the  fquare  lohich  is  made  of  AH.  VVherfore 
the  re&angle  figure  comprehended  "Under  the  lines  AB  and  BH  is  equall  to 
the  fquare  lohich  is  made  of the  line  H  A .  VVherefore  the  right  line  geuen  AB 
is  deuidedin  the  point  H, in fuel? fort  that  the  re&angle  figure  contayned  Under 
AB  and  B  H  is  equall  to  the  fquare  lohich  is  made  of  AH;  lohich  loas  required 
to  be  done.  ' 

Thys  propofition  hath  many  fingular  vfes.  Vpon  it  dependeth  the  demonftration  Many  and 
of  that  worthy  Probleme  the  10.  Propofition  of  the  4.booke  :  which  teacheth  to  de-  fingultr  vfes 
feribe  an  Ifofceles  triangle, in  which  eyther  of  the  angles  at  the  bafe  (hall  be  double  to  of  this  props- 

the  angle  at  the  toppe .  Many  and  diuers  vfes  ofa  line  fodeuided  Hull  yon  finde  in  the  fiiion. 

ij.booke  of  £uchde. 

Thys  is  to  be  noted  that  thys  Propofition  can  not  as  the  former  Propofitions  rI . 
of  thys  fe'cond  booke  be  reduced  vnto  numbers  .  For  the  line  EB  hathvnto  the  ■ 

fine  AE  no  proportion  that  can  be  named,  and  therefore  it  can  not  be  exprefled  VenducedU 
by  numbers.  For  forafmuch  as  the  fquare  qf  EB  is  equall  to  the  two  fquaresof  to  numbers  * 
AB  and  AE  (by  the  47.ofthe  firfi)  and  AE  is  the  halfe  of  A  B,  therefore  the 
line  BE  is irrationall .  Foreuenas  two  equall  fquare  numbers ioyned  together 
can  not  make  a  fquare  number :  fo  alfo  two  fquare  numbers,  of  which  the  one  is 
the  fquare  of  the  halfe  roote  of  the  other,  can  not  make  a  fquare  number .  As  by 
an  example.  Take  the  fquare  of  8.  which  is  64.  which  doubled,  that  is,  128. mi 
keth  not  a  fquare  number  •  So  take  the  halfe  of  8 .  which  is  4*  And  the  fquares  of 
8 .  and  4.  which  are  64.  and  1 6.  added  together  Hkewyfe  make  not  a  fquare  num¬ 
ber  .  For  they  make  8  o.  who  hath  no  roote  fquare .  Which  thyng  muft  ofnecelii* 
tic  be  if  thys  Probleme  fliould  haue  place  in  numbers. 

But  in  Irrationall  numbers  it  is  true3  and  may  by  thys  example  be  declared. 

X.ii.  Lee 


r 


Let  8,be  fo  deuided,that  that  which  is  produced  of  the  whole  into  one  of  his  partes 
Ihall  be  equall  to  the  fquare  number  produced  of  the  other  part.  Multiply  8. into  him 
felfe  and  there  (hall  be  produced  64.  that  is, the  fquare  CD.Deuidc  8.  into  two 

equall  partes, that is,into  4,  and  4.  as  the  line  E  oxEC .  And  multiply  4. into  hym 

felfe,and  there  is  produced  1  <5,  which  adde  vnto  64,  and  therefhall  be  produced  80: 
whoferooteis  Vg^  80:  which  is  the  line  £  2?  or  the  line  E  F  by  the  47,  of  the  firft*  And 
forafmuch  as  the  line  EFisdif  80,  &  the  lyne  E  A\s  4*  therfore  the  lynctAt Fis 
80— 4,And  fo  much  (hall  the  line  AH  be.  And  the  line  B  H  (hall  be  8 — */§"*  80— 4,that 
is  j 1 2 — Vfr  80.  Now  the  12  —  8 o  multiplied  into  8  dial  be  as  much  as  8  o — 4, 

multiplied  into  it  felfe. For  of  either  of  them  is  produced  p  6 — y/  5120. 


he  uffheorme .  The  izfPropofition . 

In  ohtufeangle  triangles,  the fquare  which  is  made  of  the fide 
fubtending  the  obtufe  angle, is  greater  then  the fquare s  which 
are  made  ofthefides  which  comprehend  the  obtufe  angle ,  by 
the  reB angle  figure,  which  is  comprehended  twife  vnder  one 
ofthofe  ftdes  which  are  about  the  obtufe  angle  ,ypon  which 
being  producedfalleth  a  perpendicular  line  3  and  that  which 
is  outwardly  taken  betwene  the  perpendicular  line  and  the 
obtufe  angle. 


Demonstra¬ 

tion. 


$ 


V^ppofe  that  ABC  he  an  ohtufeangle  triangle  bauing 
the  angle  B  AC  obtufe ,and from  the  point  B(by  the  12* 
of  the firft)dr aw  a  perpendicular  line  lonto  CA  produced 
and  let  the  fame  be  BID.  Then  I  fay  that  the  fquare 
1 vhich  is  made  of the fide  B  C3  is  greater  then  the fquares 
"Which  are  made  of the fides  B  A  and  A  C,hy  the  re  hi  am 

_ _ fff  gle figure  comprehended  lender  the  lines  CA  and  AD 

twife .  For  forafmuch  as  the  right  line  CD  is 
by  chaunce  deuided  in  the poynt  A }  therefore 
(by  the  4.  of the fccond  )  the  fquare  "Which  is 
made  of  CD  is  equall  to  the  fquares  "Which  are 
made  of  C  A  and  A  D,and  lonto  the  reflangk 
figure  contayned  Wilder  CA  and  AD  twife. 

But  the  fquare  "Which  is  made  of  DB  com * 
mon  Ivnto  them  both  .  Wherefore  the  fquares 
-which  are  made  of  CD  and  D  B  are  equall 
to  the  fquares ' which  are  made  of  the  lines  C  A, 

A  Dj  and  D  Byand  nto  the  red  angle  figure 

contayned  lender  the  lines  CA  and  AD  twife .But  (by  the  the firflfihe 

fquare  yohich  is  made  of  CB  is  equall  to  the  fquares  lehtch  are  made  of the  lines 


• '  if  y. 

■ 


■M 


ofEuclides  Elementes . 


Foi  77* 


CD  and  /D/8.  For  the  angle  at  the  point  D  is  a  right  angle .  AndWnto  the 
fquares  lohich  are  made  of  AID  and  D  H  (by  the  felfe  fame )  is  equall  the 
fquare  t/hich  is  made  of  A  /8 .  VVherfore  thefquare  H vhich  is  made  of  CHys  e- 
quail  to  the  fquares  which  are  made  of  CA  and  A  <B  and  "Onto  the  reElanglef 
gure  contayned  loader  the  lines  C  A  and  A  ID  twife.  Wherfore  j  fquare  1 vhich 
is  made  of  C/S fits greater  then  the fquares  which  are  made  of  CA  and  AD  by 
the  rectangle figure  contayned  Tender  the  lines  CA  and  AD  twife.  In  obtufe- 
angle  triangles  therefore yhe  fquare  which  is  made  of  the  fide fiubtending  the  ob * 
tufe  angle ,is greater  then  the  fquares  which  are  made  of  the  fides  YOhich  com * 
prehend  the  obtufe  angle  Jay  the  reBangle  figure  VOhich  is  comprehended  twife 
louder  one  of thofe fides  which  are  about  the  obtufe  angle gopon  which  being  pro* 
duced falleih  a perpendiculer  lyne^and  that  which  is  outwardly  taken  betwene 
the  perpendiculer  lyne  and  the  obtufe  angle :  which  was  required  to  be  demons 
Jirated. 

Of  what  force  thys  Propofition,  and  the  Propofition  following,  touching  the 
meafuring  of  the  obtufeangle  triangle  and  the  acuteangle  triangle,  with  the  ayde 
of  the  47.  Propofition  of  the  foil  booke  touching  the  rightangle  triangle,  he  fhall 
well  perceaue,which  fiiall  at  any  time  neede  the  arte  of  triangles  in  which  by  thre 
thinges  knowen  is  euer  iearched  out  three  other  thinges  vnknowen,by  helpeof 
the  table  of  arkes  and  cordes. 

The  n.Tbeoreme.  fhc  13  fPropofition. 


5^  h  acuteangle  triangles 3the fquare  which  is  made  of  the 
fide  that fubtendeth  the  acute  angle js  lejfe  then  the  fquares 
which  are  made  of  the  fides  which  comprehend  the  acute  an¬ 
gle  >b  the  reBangle  figure  which  is  coprehended  twife  vnder 
one  of thofe  fides  which  are  about  the  acuteangle ,  vpo  which 
fillet h  a  perpendiculer  lyne ,  and  that  which  is  inwardly  ta~ 
ken  betwene  the perpendiculer  lyne  and  the  acute  angle . 


■  Vpp°fe  that  ADC  be  an  acuteangle  triangle  ha* 
the  angle  aty point  D  acute  ys( by  the  iz.of 
L_ns!  t  hefirU  from  the  point  A  dr aw  "Onto  the  lyne  D 
C  a perpendiculer  lyne  JD.  Thenlfaythat  thefquare 
lohich  is  made  of  the  lyne  JC  is  lefie  then  the  fquares 
^hich  are  made  of  the  lyne  CDandD  Ajby  the  reBangle 
figure  conteyned  bonder  the  lines  CD  and  HD  twife.  For 
forafmuch  as  the  right  lyne  <B  C  is  by  chaunce  deuided  in 
the  point  Dyher fore  (by  the  7.  of the fiecond)  the fiq  uares 

X.iij.  which 


A 


DemnJlratiS 


The  fecond  Boothe 

lohich  are  made  of the  lines  C  B  and  B  D  are  e quail  to  the  reBangle  figure  com 
tamed  lander  the  lines  C  B  a?id  D  B  twife  and  Tanto  the  fquare  lohiche  ismade 
of  line  C  D  .Tut  the  fquare  lohich  ismade  of  the  line  !D  A  common  lanto  them 
both .  VVherfore  the  fquares  lohich  are  made  of  the  lines  CB  ,  BD,and(D 
jf3are  equall  lanto  the  re  El  angle  figure  contayned  lander 
the  lines  C  B  and  B  ID  twife ,  and  lanto  the  fquares  lohich 
are  made  of  AD  and  D  C.  But  to  the fquares  lohiche  are 
made  of  the  lines  B  D  and  D  A  is  equal y  fquare  lohich  is 
made  of  the  line  JIB :  for  tl/ angle  aty  point  D  is  a  right 
angle .  Andlanto  the  jquares  lohiche  are  made  of  the  lines 
A  D  and  D  C  is  equall  the  fquare  lohiche  is  made  of  the 
line  A  C(  by  the  a-.ofy  firfi):loberfore  the  fquares  lohich 
are  made  of  the  lines  C  B  and  B  A  are  equal  to  the fquare 
lohich  is  made  of  the  line  A  C,and  to  that  lohich  is  contain 
7ied louder  the  lines  C Band BD  twife  .  Wherfiorethe  b 
fquare  lohich  is  made  of  the  line  A  C  beyng  taken  alone  js  lejfe  then  the fquares 
lohich  are  made  of  the  lines  C  B  and  BA  by  the  rectangle  figure  ,  lohich  is  con * 
tainedlonder  the  lines  C  B  and B  D  twife .  In  rectangle  triangles  therfore  the 
fquare  lohich  is  made  of  the  fide  that  fubtendeth  the  acute  angle, is  lejfe  then  the 
Jquares  lohich  are  made  of  the  fides  lohich  comprehend  the  acute  angle  ,  by  the 
rectangle  figure  lohich  is  comprehended  twife  louder  one  of thofie fides  lohich  are 
about  the  acute  angle,  lopon  lohich falleth  a  perpendicular  line  ,  and  that  lohich 
is  inwardly  taken  betwene  the  perpendicular  line  and  the  acute  angklehich  loas 
required  to  be proued. 


1]  jt  Corollary  added  by  Orontius. 


A  Corollary. 


This  Propo/i- 
tion  true  in  all 
kindes  of 
triangles * 


Hereby  is  eafily  gathered,that  fuch  a  perpendicular  line  in  redangie  triangles 
falleth  ofneceffitie  vpon  the  fide  of  the  triangle,  that  is,  neyther  within  the  trian¬ 
gle, nor  without.  But  in  obtufeangle  triangles  it  falleth  without, and  in  acuteangle 
triangles  within .  For  the  perpendicular  line  in  obtufeangle  triangles,  and  acute- 
angle  triangles  can  not  exa&ly  agree  with  the  fide  of  the  triangle :  for  then  an  ob- 
tufe  &  an  acuteangle  ihould  be  equal  to  a  right  angle,contrary  to  the  eleuenth  and 
twelfth  definitions  of  the  firft  booke .  Likewife  in  obtufeangle  triangles  it  can  not 
fall  within, nor  in  acuteangle  triangles  without:  for  then  the  outward  angle  of  a 
triangle  fhould  be  lefte  then  the  inward  and  oppofite  angle,whichis  contrary  to 
the  i  ^.of  the  firft. 

And  this  is  to  be  noted, that  although  properly  an  acuteangle  triangle,  by  the 
definition  therof  geue  in  the  firft  booke,be  that  triangle, whofe  angles  be  all  acute: 
yet  forafinuch  as  there  is  no  triangle,but  that  it  hath  an  acute  angle,this  propofith 
on  is  to  be  vnderftanded,&  is  true  generally  in  all  kindes  of  triangles  whatfoeuers 
and  may  be  declared  by  them, as  you  may  eafily  proue. 

The 


of Euclides  Elements s.  FoL  80, 

The  lEProbleme.  Thev^.Tropofition. 

Vnto  a  reBiline figure  geuen3to  make  a jquare  squall. 


^The  ende  of the  fecond  Booke 

ofEuclides  Elementes. 


Cofijiruftm. 


Dmo»tJlraii$ 


The  Argument 
of  this  kooks* 


The  fir  ft  defi¬ 
nition* 


fflby  circlet 
take  their 
equality  of 
their  diame- 
ten  or  femi¬ 
diameters • 


i[.The  third  books  of  Eu- 

elides  Elementes. 

His  third  booke  ofEuclide  entreateth 

of  the  moft  perfed:  figure,  which  is  a  circle .  Where¬ 
fore  it  is  much  more  to  be  eftemed  then  the  two 
bookes  goyng  before ,  in  which  he  did  fet  forth  the 
moft  fimple  proprieties  of  rightlined  figures  .  For 
fciences  take  their  dignities  of  the  worthynes  of  the 
matter  that  they  entreat  of.But  of  al  figures  the  circle 
is  of  moft  abfoluteperfedion,whole  proprieties  and 
pafsions  are  here  fet  forth,and  moft  certainely  demo- 
ftrated.Here  alfo  is  entreated  ofrightlines  fubten- 
ded  to  arke's  in  circles  :  alfo  of  angles  fet  both  at  the 
circumference  and  at  the  centre  of  a  circle,andofthevarietie  and  differences  of 
them.Wherfore  the  readyng  of  this  booke ,  is  very  profitable  to  the  attayning  to 
the  knowledge  of  chordes  and  arkes.lt  teacheth  moreouer  which  are  circles  con- 
tinget,and  which  are  cutting  the  one  the  other  :  and  alfo  that  the  angle  of  contin- 
gence  is  the  leaft  of  all  acute  rightlined  angles:and  that  the  diameter  in  a  circle  is 
the  longeft  line  that  can  be  drawen  in  a  circle .  Farther  in  it  may  we  learne  how, 
three  pointes  beyng  geuen  how  foeuer(fo  that  they  be  not  fet  in  a  right  line), may 
be  drawen  a  circle  palling  by  them  all  three,  Agayne, how  in  a  lolide  body ,  as  in  a 
Sphere,Cube,orfuchlyke,maybe  found  the  two  oppofite  pointes .  Whiche  is  a 
thyngvery  necefiaryand  commodious  :  chiefly  for  thofe  that  fliall  make  inftru- 
mentes  feruyng  to  Aftronomy,and  other  artes. 

! Definitions . 


Equal!  circles  are  fuch^hofe  diameters  are  equally  or  who/e 
lynes  drawen  from  the  centres  are  equal! . 


The  circles  A  and  B  are  equal, if  theyr  diameters,namely,E  F  and  C  D  be  equaShor  if 
their  femidiameters  ,  whiche  are  lynes  drawen  from  the  center  to  the  circumference^ 
namely  A  F  and  B  D  be  equall. 


i  The  reafon  why  circles 
take  theyr  equalitie ,  of  the  e- 
qualitie  of  their  diameters  or 
femidiameters  is  ,  for  that  a 
circle  is  delcribed  by  one  re- 
uolution  or  turnyng  about  of 
the  lemidiameter,  hauing  one 
of  his  endes  fixed .  As  if  you 
Imagine  the  lyne  A  E  to  haue 

his  one  point  namely  A  faftened,and  the  other  end  namely  E  to 

it 


o/Sudides  Elementes.  Fol.2u 

it  come  to  the  place  where  itbega  to  moue ,  it  fhal  fully  defcribe  the  whole  circle. 
Wherefore  ifthefemidiameters  bee  equall5the  circles  of neceffity e  muft  alfo  be 
equal!:  and  alfo  the  diameters. 


By  thys  alfo  is  kno wen  the  definition  of  vnequall  circles.  Definition  of 

imeqHtUw* 

Circlestyhofe  diameters  orfmidiameters  are  vneqnatt,  are  alfo  vneqnai  .  *And  that  circle  Wft 
fybicb  hath  the  greater  diameter  or  femidiametcr,  is  the  greater  circle »  and  that  emit  Which  hath 
the  lejfe  diameter  or  femidiamet  ert  it  the  leffe  circle „ 


As  the  circle  t  M  is  greater 
then  the  circle  I K ,  for  that  the 
diameter  L  M  is  greater  then  the 
diameter  I K:  or  for  that  the  femi- 
diameterG  Lis  greater  then  the 
femidiameterHI. 


right  line  is  fay  d to  touch  a  circle jtohich  touching  the  cir > 
ele  and  being produced  cutteth  it  not .  mm* 


As  the  right  lyneEFdrawen  from  the  point  E  9andpa%ng  by  a  point  of  the  circle* 
namely ,by  the  point  G  to  the  point .F  on¬ 
ly  toucheth  the  circle  G  H,and  cutteth  it 
not, nor  entreth  within  it.For  a  right  line 
entryng  within  a  circle, cutteth  anddeui- 
deth  the  circle .  As  the  right  lyne  K  L  de¬ 
ni  deth  and  cutteth  the  circle  K  L  M ,  and 
entreth  within  it :  and  therfore  toucheth 
it  in  two  places  .  But  a  right  lync  tou- 
chyng a  circle, which  is  commonly  called 
a  cotingent  lyne,toucheth  the  circle  one* 
ly  in  one  point. 


df  contigent 
tine* 


% 


Circles  are  fayd to  touch  the  one  the  other ythicb  touching  the  V’M&foh 
one  the  other  jut  not  the  one  the  other*  ***** 


As  the  two  circles  AB  and  BC  touch  the 
one  the  other  ♦  For  theyr  circumferences 
touch  together  in  the  poynt  B  *  But  neither  of 
them  cutteth  or  deuideth,  the  other .  Neither 
doth  any  part  of  the  one  enter  within  theo« 
ther.And  fuch  a  touch  of  circles  is  euer  in  one 
poynt  onely:  yhich  poynt  oncly  is  common 
to  them  both  .As  the  poynt  B  is  fit  the  confc* 
tehee  of  the  circle  A  B,and  aUb  is  io,  the  circu- 
ifereace  of  the  circle  BC. 


The  touch  of 
emits  is  tuev 
in  one  point 
mdj.  , 

Aa-f.  Circles 


Circles  mg 
South  toge¬ 
ther  two  md- 
mrofurdjes. 


Fourth  defi- 
nuion. 


Fife  defini¬ 
tion. 


touch  together  two  matier  of  wayes ,  either  outwai^ly  &£qij,^ 
wholy  without  the  other :  or  els  the  one  being  cojitayne4  withiuthe  otfc. 

As  the  circles  D  E  and  D  F :  of  which  the  one  D  E  contay- 
neth  the  other ,  namely  D  F :  and  touch  the  one  the  other  in 
the  poynt  Z>;and  that  oncly  poynt  is  common  to  them  both : 
neitherdoththeone  enter  into  the  other  »Ifanypart  ofthe 
pne  enter  into  any, part  of  the  other,then  the  one  cutteth  and 
feideth  the  other  ,  and  toucheth  th?  one  the  other  not  in 
one  poynt  oneiy  as  in  the  other  before ,  blit  in  two  pbintes, 
and  haue  alfo  a  fupcrficies  common  to  them  both.  As  the  cir¬ 
cles  G  H  ifand  HL  K  cut  the  one  the  other  in  two  poyntes 
H and  X;and  the  one  entreth  into  the  other :  Al¬ 
fo  the  fuperficies  AfiC  is  common  to  them  both; 

For  it  is  a  part  of  the  circle  G  H  K ,  and  alfo  it  is  a 
part  pf  the  circle  HL  K. 


%ight  lines  in  a  circle  are  fay  d  to 
be  equally  diBant  from  the  cen~ 
freshen  perpendicular  lines  drawen  from  the  centre  ynto 
thoje  lines  are  equall,  <*And that  lint  is  fayd  to  be  more  di~ 
Hantfvpon  wbomfalleth  the  greater  perpendicular  line. 


As  in  the  circle  tABCDw hofe  centre  is  E,  the  two  lynes 
JI  B  and  CD haue  equall  difiance  from  the  centre  E  .*  bycaufe 
that  the  Iyne  EF  drawen  from  the  centre  E  perpendicularly 
vpon  the  lyne  A  2?,and  the  lyne  E  G  drawen  likewife  perpendi- 
larly  from  the  centre  E  vpon  the  lyne  CD  are  equall  the  one  to 
theother  .  Butin  the  citdzH  KLM  whofc  centreis  2\^the 
lyne  H  if  hath  greater  diftance  from  the  centre  2^  then  hath 
the  lyne  L  M :  for  that  the  lyne  O  2^drawen  from  the  centre 
^perpendicularly  vppon  the  lyne  HKis  greater  then  the  lyne 
T^JF  which  is  drawen  fro  the  centre  perpendicularly  vpon 
the  lyne  L 

So  likewife  in  the  other  figure  the  lynes  AB  and  D  Cin  the 
circled  BCD  are  equidiftant  from  the  centre  C,bycaufe  the 
iyncsOCandiy  jP  perpendicularly  drawen  from  the  centre  G 
vppon  the  fayd  lynes  A  B  and  D  Care  equall  .  And  the  lyne 
A  B  hath  greater  diftance  from  the  centre  G  then  hath  the 
the  lyne  EF  ,  bycaufethclync  O  G  perpendicularly  drawen 
from  the  centre  G  to  the  lyne  cAB  is  greater  then  the  lyne  H 
G  whiche  is  perpendicularly  drawen  from  the  centre  G  to  the 
lyneEF, 


A feBion  or figment  of  a  circle 9  is  a  figure  coprehendedmder 
a  right  line  and  a  portion  ofthe  circumference  of  a  circle. 


As  the 


As  the  figure  ABCis  a  fedtion  of  a  -circle 
bycaufe  itis  comprehended  vnder  the  right 
lyne  AC  and  the  circumference  of  a  circle  A. 

B  C .  Likewife  the  figure  D  E  Ais  a  fedtion  of 
a  circle  ,  for  that  it  is  comprehended  vnder 
the  right  Ivne  D  F ,  and  the  circuference  D  E 
F.  And  the  figure  A  BC  for  that  it  cotaineth 
within  it  the  centre  of  the  circle  is  called  the 

greater  fection  of  a  circle :  and  the  figu  re  CD  E  Fis  the  lefle  fection  of  a 
itis  w holy  without  the  centre  of  the  circle  as  it  was  noted  in  the  1 6 .  Definition  ortne 
firft  boofce* . ..  f  - 


*tAn  angle  ofafeBion  or  fegment ?  is  that  angle  which  is  con  < 
tajneciyncier  a  right  line  and  the  circuference  of  the  circle . 

As  the  angle  A  B  C  in  the  fedtion  A  B  C  is  an  angle  of  a  fee- 
tiort ,  bycaufe  it  is  c6ntained  of  the  circumference  B  A  C  and 
the  right  lyne  B  C  .  Likewife  the  angle  C  B  D  is  an  angle  of  the  ^ 
fedtion  B  D  C  bycaufe  it  is  contayned  vnder  the  circumference 
B  D  C,andthe  right  JyneB  C .  And thefe  angles  are  commonly  I 
called  mixte  angles,  bycaufe  they  are  contayned  vnder  a  right  ' 
lyne  and  a  crooked  .  And  thefe  portions  of  circumferences  are 
commonly  called  arkes,  and  the  right  lynes  arc  called  chordes, 
or  right  lynes  fubtended.  And  the  greater  fedtion  hath  euer  the 
greater  angle,and  theleffe  fedtion  the  Idle  angle, 

Jn  angle  is  fayd  to  be  in  afeBionjtohein  the  circumference  is 
takgn  any  poynt^andfrom  thatpqynt  are  drawen  right  lines 
to  the  endes  of  the  right  line  which  is  the  bafe  of  the  fegment ? 
the  angle  which  is  contayned  vnder  the  right  lines  drawen 
from  the pojnt,  is  ( Ifayfaydto  be  an  angle  in  a JeBion . 

As  the  angle  A  BCfs  an  anglein  the  fedtion  A  B  C ,  bycaufe 
from  the  poy  nt  B  beyng  a  poynt  in  the  circumference  A  B  C  are 
drawen  two  right  lynes  B  C  and  B  A  to  the  endes  of  the  lyne  A  G 
which  is  the  bafe  of  the  fedtion  A  B  C  .  Likewife  the  angle  ADC 
is  an  angle  in  the  fedtion  A  D  C, bycaufe  from  the  poyrit  D  beyng 
in  the  circuference  A  D  C  are  drawen  two  right  lynes,namelv,D 
C  &  D  A  to  the  endes  of  the  righ  t  line  A  C  which  is  alfo  the  bafe 
to  the  fayd  fedtion  A  D  C-So  you  fee,it  is  not  all  one  to  fay,  an  am 
gle  of  a  fedtion,and  an  angle  in  a  fedtion.An  angle  of  a  fedtion  co- 
fifteth  of  the  touch  of  a  right  lyne  and  a  crooked.  And  an  angle 
in  a  fedtionis  placed  on  the  circumference ,  and  is  contayned  of  two  right  lynes .  Alfo 
the  greater  fedtion  hath  in it  the  leifc  angle ,  and  the  leife  fedtion  hath  in  it  the  greater 
angle.-  'v'  ■  o-.' .  " 

'‘But  when  the  right  lines  which  comprehend  the  angle  do  re - 
ceaue  any  circumference  of  a  circle jben  that  angle  is  fayd  to 
.  he  correfgondent^and  to pertaine  to  that  circumference* 

Aa.i/,  As  the 


Sixt  defm~ 
tion. 


Mfot  Angles, 

Arkes. 

C hordes , 

Seuenth 
finitim «•. 


Difference  of 
an  angle  of  £ 
Seffion.and 
of  an  an  Ae 
in  a  Semon» 


Eight  deft 
nition » 


plinth  dtfi- 
mtion . 


Tenth  dcfini - 
tion. 

T wo  defini¬ 
tions, 

Brft, 


Semi* 


As  the  right  lynes B  A  ~ 

€,and  reeeaue  the  circumference  A  D  C  iherforc  the  angle  A  B 
C  is  fayd  to  fubtend  and  to  pettaine  to  the  circuference  ADC. 
And  if  the  right  lynes  whiche.  caufe  the  angle,  concurrein  the 
centre  of  a  circle :  then  the  angle  is  fayd  to  be  in  the  centre  of  a 
circle « As  the  angle  E  F  D  is  fayd  to  be  in  the  centre  of  a  circle, 
for  that  it  is  comprehended  of  two  right  lynes  F  E  and  F  Ds 
rhiehe  concurre  and  touch  in  the  centre  F.  And  this  angle  likewife 
fubtendeth  the  circumference  EG  D  :  whiche  circumference  alfo, 
is  the  meafure  of  the  greatnes  of  the  angle  E  F  D* 


'  of  a  circle  is  (an  angle  being  fetal  the  e 

centreof  a  circle )  a  figure  contayned  ynder  the  right  lines 
which  make  that  angle 9and  the part  of  the  circumference  re* 
ceaucd  ofthem. 


As  the  figure  A  B  C  is  a  fedor  of  a  circle,  for  that  it  hath  an  angle 
at  the  centre,namely  the  angle  B  A  C,&  is  cotained  of  the  two  right  j 
lynes  A  B  and  A  C  (  whiche  contayne  that  angle  and  thedreumfe. 
rence  receaued  by  them. 


Likefegmentes  orfellions  of  a  circle  are  thofe9  which  ham 
equall  angles %or  in  whom  are  equall angles • 


Here  are  fettwo  definitions  of  like  fedions  of 
a  circle. The  one  pertaineth  to  the  angles  whiche 
are  fet  in  the  centre  of  the  circle  and  reeeaue  the 
circumfercce  of  the  fayd  fedions:  the  other  per¬ 
taineth  to  the  angle  in  the  fedion,  whiche  as  be¬ 
fore  was  fayd  is  euer  in  the  circumference  •  As  if 
the  angle  B  AC,  beyng  in  the  centre  A  and  re- 
ceauedof  the  circumference  B  LC  be  equall  to 
theangleFEG  beyng  alfointhe  centre  E  and 
receaued  of  the  cir  cu  inference  F  KG,  then  are  the  two  fedions  B  CL  and  FGKIyfc® 
by  the  firft  definition. By  the  fame  definition  alfo  are  the  other  two  fedions  like3naae* 
ly  B  C  D,and  F  G  H/or  that  the  angle  BAG  is  equall  to  the  angle  F  E  G. 


Alfo  by  thefecond  definition  if  B 
A  C  beyng  an  angle  placed  in  the  Cir¬ 
cumference  of  the  fedion  B  C  A  be  e- 
angle  E  D  F  beyng  an  angle  in  the  fe¬ 
dion  E  F  D  placed  in  the  circumfe¬ 
rence,  there  are  the  two  feCtionsBC 
A ,  and  E  F  D'lyke  the  one  to  the  o- 
ther .  Likewife  alfo  if  the  angle  BGC 
beyng  in  the  fection  B  C  G  be  equall 
to  the  angle  E  H  F  beyng  in  the  fedio 
.  E  H  F  the  two  fedions  BCG  and  E  F 
H  are  lyke.  And  fo  is  it  of  angles  beyng  equall  in  any.poynt  of  the  circumference. 

,  \  v  -  -  Bucllde* 


Euclide  defmcth  not  equall  Se&ions  :for  they  may  infinite  wayes  be  defcribed. 
For  there  may  vpponvnequall  right  lynes  he  fet  equall  Se&ions  (butyet  in  vne- 
quall  circles)  For  from  any  circle  beyng  the  greater,may  be  cut  of  a  portion  equall 
to  a  portion  of  an  other  circle  beyng  the  lelfe .  But  when  the  Sections  are  equall, 
and  are  fet  vpon  equall  right  lynes ,  theyr  circumferences  alfo  flialbe  equall  *  And 
right  lynes  beyng  deuidedinto  two  equall  partes ,  peroendicular  Ivnes  drawer* 
from  the  poyntes  of  the  diuifion  to  the  cir- 
cumfercces  ihalbe  equall.  A  s  if  the  two  fc£ti- 
oiis  ABC and  CD  E  F,  beyng  fet  vppon  equall 
ryghtlyn.es  ACdcDF,  be  equall :  then  if  eeh  of 
the  two  lynes  ACgcDF  be  deuidedinto  twoe- 
quall  partes  in  the  poyntes  G  and  //*,&  from  the 
uyd  poyntes  be  drawento  the  circumferences 
two  perpendicular  lynes  BG  and  EH,  the  fayd 
perpendicular  lynes  fhalbe  equall ♦ 


ZfofThe  ifProbteme .  The  i.  Tropofition . 


To finde  out  the  centre  of  a  circle geuen. 


Vppofe  that  there  lea  circle  geuen  ABC .  It  is  requi¬ 
red  to  finde  out  the  centre  of the  circle  ABC.  (Draw  in  it 
aright  line  at  all  aduenturesy  and  let  the fame  he  A  B. 
And  (by  the  i  o*  of  the  firf)  deuide  the  line  A  B  into  two 
equall partes  in  the poynt  1 ).  And(by  the  i  i.of  the  fame ) 
|  fro  the poynt  D  raife  yp  lento  AS  a  perpendicular  line 

_  _ |  D  by  the  fecond petition  )extend D  C lento y  point 

E.  And( by  the  io*of  the firft )  deuide 
the  line  C  E  into  two  equall  partes  ' 
the  poynt  F .  T  hen  1 fay  that  the  point 
E  is  the  centre  of  the  circle  ABC.  For 
if it  be  not  fet fome  other  po 
Gjbe  the  centre.  And  (by  th 
tion)draw  thefe  right  lines 
and  GB.  And  forafmuch 
equall  lento  D  By  and  ID  G  is 
lento  the  both,  therefore  thefe  two  lines 
A  D  and  DG  are  equall  to  thefe  two 
lines  G  D  and  D  B?the  one  to  the  other  find  (  hy  the  .  definition  of the  firf) 

the  bafe  G  jC  is  equall  to  the  bafe  G  B .  For  they  are  both  drawen  from  the  ceri* 
tre  G  to  the  circumference :  therefore  (  hy  the  8.  of the  frft )  the  angle  ADO 
is  equall  to  the  angle  BDG .  But  ~%>hen  a  right  line Jlandmg  lepon  a  right  line 
maketh  the  angles  on  eche  fide  equall  the  one  to  the  other  y  eyther  of  thofe  angles 
( by  the  io*  definition  of the firjl)  is  a  right  angle.  WJeerefore  the  angleBDG 

yfa.iij.  is  4 


Why  SaetlM 

defineih  not 
equalised 

cm* 


CouflruZlkn* 


tion  leading 
to  an  impofL 
fibilitk* 


€meUiy* 


lOrnon/lra* 
uo  leading  to 
an  intpoffibi- 
Stie. 


The  thirdflooke 

is  a  right  angle  :  butj  angle  FID  ©  is  alfo  a  right  angle  by  confiruBion.  Vtfher* 
fore  (by  the  4 .  petition)the  angle  FT)  'Bis  e  quail  to  the  angle  ©  T)  Gythe  grea¬ 
ter  to  the  lejfe ,  K>hich  is  impofhble .  Wherefore  the poynt  G  is  not  the  centre  of 
the  circle  ABC,  In  like  K>ife  may  ~fre  prone  that  no  other  poynt  befides  F  is  the 
centre  of  the  circle  AFC.  Wherefore  the  poynt  F  is  the  centre  of  the  circle 
AT  C:  lohkh  leas  required  to  be  done . 

Cor r el  ary* 

Hereby  it  is  manifefljhat  if  in  a  circle  a  right  line  do  deuide 
a  right  line  into  two  e quail partes, and  make  right  angles  oneche 
fide:  in  that  right  line  which  deuideth  the  other  line  into  two  e* 
quail  partes  is  the  centre  of  the  circle* 

*r&The  1, T heoreme .  The  zfPropofition. 


If  in  the  circuference  of a  circle  be  taff  two poyntes  at  all  ad- 
uentures :  a  right  line  drawen from  the  one poynt  to  the  other 
Jhall  fall  within  the  circle. 

Vppofe  that  there  be  a  circle  A  ©  C.  And  in  the  circumference  tier* 
of  let  there  be  take  at  all  adnentures  thefe  two  poyntes  A  &  ©.  Then 
I fay  that  aright line  dr awen from  AtoT  Jhall fall  loithin  the  circle 
'A  ©  C.For  if  it  do  not  Jet  it  fall  without  the  circle, as  the  line  AFT 
dotbptohich  if it  be  pofiible  imagine  to  be  a  right  line .  And(  by  the  Tropojition 
going  before)take  the  centre  of  the  circle, and  let  the  fame  be  D.Andfby  the firfi 
petitwn)draw  lines from  ID  to  A, and from  ID  to  ©„  And  extend  D  F  to  E*  And 
for  afmuch  as  (by  the  15*  definition  of y  firfi ) 

T)  A  is  equall  lonto  DT.  T  here  fore  the  an* 
gle  D  A  Eis  equall  to  the  angle  ID  ©  E.And 
for  afmuch  as  o?ie  of  the  fides  of  the  triangle 
(DAE,  namely  the  fide  A  ET  is  produced, 
therefore  (  by  the  16  ♦ of  the  firfi  )  the  angle 
(D  E  T, is  greater  then  the  angle  D  A  EJBut 
the  angle  DA  E  is  equall  ynto  the  angle 
DTE.  Wherfore  the  angle  DET  is  great 
ter  then  the  angle  DTE.  Tut  (by  the  iS. of 
the  firfi)  Tmto  the  greater  angle  is  fubt ended 
the  greater fide .  Wherefore  the fide  DTis 
-greater  then  the  fide  D  K  Tut(by  the  15  ♦  definition  of the  firfi )the  line  D  Tjs 

equall 


0. 


FoLty, 

equaU  lento  the  line  0F.  VVhcrfore  the  line  • D  F  is  greater  then  the  lint  0  Ey 
namely y  the  lejfe greater  then  the greater Ytohtch  is  irnpofiibk.Whtrforea  right 
line  drawenfrom  AtoB  falleth  not  without  the  circle.  In  like  fort  alfo  may 
prone  that  it  falleth  not  in  the  circumference :  Wherefore  it  falleth  within  the  , 
circle .  If  there  fore  in  the  circumference  of a  circle  he  taken  two  poyntes  at  dll  ad* 
uentures:  a  right  line  drawenfrom  the  onepoynt  to  the  other frail fall  1 whin  the 
circle:  U? hid:  t>as  required  to  he proued . 

y&The  zfiTheoreme.  The  ifPropofmon. 

If  in  a  circle  a  right  line pafiing  by  the  centre  do  deuide  ano¬ 
ther  right  line  not  pa  fling  by  the  cetre  into  two  equall  partes: 
it Jh all  deuide  it  by  right  angles .  And  if  it  deuide  the  line  by 
right  angles fit fh all  alfo  deuide  the fame  line  into  two  e quail 
partes . 

Vpjmfe  that  there  he  a  circle  A  B  Cy  and  let  there  he  in  it  drawen  ybe&flHrt 
a  right  line  pafiing  by  thexentye,  and  let  the fame  he  C0>  deuiding  of  this  Aopo 
ah  other  tight  line  A  B  not  pafiing  by y  centre  into  two  e  quail  partes 
in  the poynt  F.  T hen  I  fay  that  the  angles  at  the poynt  of  the  deuijion 
are  right  angles.  T ahe  ( hyy  firfl  of  the  third) 
the  centre  of  the  circle  A B  C\  and  let  the fame 
he  E .  And(  by  the  firft  petition )  drawe  lines 
from  E  to  Ayr from  E  to  B>  And  for  afmuch 
as  the  line  A  F is  equall lento  the  line  F'Byand 
the  line  F  E  is  common  to  them  bothy  therfore 
thefe  two  lines  EF  and  FA  are  equall  lento 
thefe  two  lines  E  F&  F  B.  And  the  hafe  E  A 
is  equall  lento  the  hafe  E  B  (by  the  15.  defnk 
tion  of  thefirfi) .  Wherefore( by  the  8.  of  the 
firft) the  angle  AFE  is  equall  to  the  angle 
BFE.  But  Sfrhen  a  right  line  Handing  lepon  a  right  line  doth  make  the  angles 
■on  etc he fide  equall  the  one  to  the  other yeyther  of  thofe  angles  is  (by  the  10. define 
tion  of  the  firft)a  right  angle. VVherf ore  either  of  thefe  angles  A  FEyfy  B  F  E 
is  a  right  angle.  VVherefore  the  line  C 0 pafiing  by  the  centre y  and  deuiding  the 
line  A  B  not  pafiing  by  the  centre  into  two  equall partes ymaketh  at  the  point  of 
the  deuijion  right  angles. 

But  now  fuppofe  that  the  line  C0  do  deuide  the  line  A B  in  fuch  fort  that  it  The  feccnd 
■maketh  right  angles.  Then  I  fay  that  it  deuideth  it  into  two  equall  partes  y  that 
'tsyythe  line  AFis  equal!  J>nto  the  line fiB.  For  the  fame  order  of conftruction 
remay ?iing  for  afmuch  as  the  line  E  A  is  equaU'tmto  the  line  EB(hy  the  15.  de*  til™” 

AaMij.  finition 


Conjirutiiwo 


'DtmonUra- 

tion. 


finkion  of the fir  ft),  T  herefore  tlx  angle  EAF  is  e quail  ^nto  the,  angle  E  ®  F 
( by  the  5 .  of the  fir  ft).  And  the  right  angle  A  EE  is  ( by  the  4  *  petition)  equaBk 
to  the  right  angle  EE  E.  Wherefore  there  are  two  triangles  E  AF,Z?  EEF 
hatting  two  angles  equal l  to  two  angles, £?  one  fide  equall  to  one  fide,  namely  the 
fide  E  F  "Which  is  common  to  them  botb,a?id fubtendeth  one  of  the  equall  angles , 
"Wherefore  (by  the  26.  of the firfi)the fides  remayning  of  the  one, are  equdll  y?t* 
to  the  fides  remayning  of  the  other .  Wherefore  the  line  jtF  is  equall  Wn to  the 
line  F  E .  If  therefore  in  a  circle  a  right  line  pafiing  by  the  centre  do  deu'ide  an 
other  right  line  not  pafiing  hy  the  centre  intoXwo  equall  partes  ,  it jhall  deuide  it 
by  right  angles .  And  if  it  deuide  the  line  by  right  angles  it Jhall  alfo  deuide  the 
fame  line  into  two  equall  partes :  "whick"Was  required  to,  be  demonfirated. 

Theoreme.  Ifihe  ^fPropqfition. 

If  in  a  circle  two  right  lines  not  pafiing  hy  the  centre >  deuide 
the  one  the  other  :  t hey  jh all  not  deuide  eche  one  the  other 
into  two  equall  partes* 


-Demn&ra- 
tion  leading 
to  an  mpof- 
Jtbilitie* 


Fppofe  that  there  be  a  circle  AECD,  and  let  there  be  in  it  drawen  tw§ 
fright  lines  not  pafiing  by  the  centre  and  deuidingthe  one  the  other ?  and 
--  let  the  fame  le  A  C  and  E  ID,  "Which  let  deuide  the  one  the  other  in  ths 
poynt  E.  T hen  I  fay  that  they  deuide  not  eche 
the  one  the  other  into  two  equall  partes ,  For  if 
ithepofiible  let  them  deuide  eche  the  one  the 
other  into  two  equall  partes, fo  that  let  AE  be 
equall  lento  E  €,<&•  E  E  Wnto  E  ID.  And  take 
the  centre  of  the  circle  AECD  ,  "Which  let 
be  F .  And  (by  the  firfi  petition  )  draw  a  line 
from  F  to  E  .Flow for  afinuch  as  a  certaine 
right  line  EE  pafiing  by  the  centre  deuideth 
an  other  line  A  C  not  pafiing  hy  the  centre  into 
two  equall  partes, it  maketh  "where  the  deuifi* 
on  is  right  angles  (by  the  Z*  of  the  third  ) .  VVherfore  the  angle  EE  A  is  a  right 
angle.  Againe for  afinucb  as  the  right  line  F E, pafiing  by  the  centre  j  deuideth 
the  right  line  E  D  not  pafiingby  the  centre  into  two  equall  partes  ,therefore(by 
the fame  fit  maketh  "where y  deuifion  is  right  angles.  Wherfore  the  angle  EE  E 
is  aright  angle.  And  it  is proued  that  the  angle  FE  A  is  a  right  angle.  VVher- 
fore( by  the  4  ♦  petition)  the  angle  F  E  A  is  equall  lento  the  angle  F E  E, namely 
the  lejfe  angle  lento  the  greater :  "Which  is  impofiible .  Wherefore  the  right  lines 
A  C  andED  deuide  not  eche  one  the  other  into  two  equall  partes.  Iftherforem 
a  circle  two  right  lines  not  pafiing  by  the  centre,  deuide  the  one  the  other  ,  they 

fhaft 


ofSttclides  Elementes. 


Fol.%5. 


JhaUnot  deuide  eche  one  the  other  into  two  equall partes :  'which  ~tyas  required  2ft 
he demon f  rated. 

In  this  Propofition  are  two  cafes.For  the  lines  cutting  the  one  the  other,do  ey-  yw0  Caj~es  -tn 
ther,neyther  of  them  paffe  by  the  centre, or  the  one  of  them  doth  paffe  by  the  cen-  tins  Propo- 
trc,&  the  other  not.The  firft  is  declared  by  the  author.The  fecond  is  thus  proued,  fttm. 


Suppofe  that  in  the  circle  tA  BCD  the  line 
B  CD  paffing  by  the  centre  doc  cut  the  line  tA  C 
not  palling  by  thecentre.Thenl  fay  that  the  lines 
C  and  B  D  do  not  dcuidc  the  one  the  other  in¬ 
to  two  equall  partes « For  by  the  former  Propor¬ 
tion  the  line  B  D  paffing  by  the  centre  and  deui- 
ding  the  line  aA  C  into  two  equall  partes,  it  lhall 
alfo  deuide  it  perpendicularly.And  for  afmuch  as 
the  line  A  C  deuideth  the  line  B  D  into  two  equall 
partes  &  right  angled  wife:therfore  by  the  Correb 
lary  of  the  firft  of  thys  booke,the  line  *A  C  palfeth 
by  the  centre  of  the  circle  :  which  is  cotrarytothe 
fuppofitiom  Wherfore  the  lines  eA  C  and  B  D  do 
not  deuide  the  one  the  other  into  two  equal! 
partes  :  which  was  required  to  beproued. 


ConflmBton 
for  thepmid 
safe, 

•  DemnUrfr 
Sion. 


5-fc>T he  4.0  Theorem:  The  5.  Tropofitioru 


If  two  circles  cut  the  one  the  other Jthey  heme  not  one  and  the 
fame  centre . 


\Vppofe  that  thefe  two  circles 
fBCj  and  CFG  do  cut  the 
1  one  the  other  in  the  poyntes  C 
and  <B .  Then  I  fay  that  they  haue  not 
one  is*  the fame  centre. For  if  it  be  pofii* 
hie  let  E  be  centre  to  them  both.Andfby 
the firf  petition )  draw  a  line  from  E  to 
C .  And  draw  an  other  right  line  EFT 
at  aldaduentures.And for  afmuch  as 
poyfit  E  is  the  centre  of  the  circle  AFC y 
therefore  (  by  the  1 5*  definition  of  the 
firsfthe  line  E  C  is  equall  lento  the  line 

E  F.  yfgaynefor  afmuch  as  the poynt  E  is  the  centre  of  the  circle  C  T>  G,  then# 
forefby  the fame  definition)the  line  EC  is  equall  Imto  the  line  EG.  And  it  is 
proued  that  the  line  E  C  is  equall  lento  the  line  E  F :  therefore  the  line  E  F  alfo 
is  equall  l?nto  the  line  E  Gqnamely  the  lejfe  Imto  the  greater :  lohich  is  impofil + 
ble.  VVberfore  the  poynt  E  is  not  the  centre  of  both  the  circles  A  IB  Cgs*  CFG . 
In  like fort  alfo  may  prone  that  no  other  poynt  is  the  centre  of  both  the  fayd 

circles. 


CehjlrH&ktu 


Demon  ft  ra~ 
tio  leading  to 
mmpojftbi - 
litis  a 

.  Jm* 


9. 


Dmwftra- 
Sion  leading 
to  an  impQ]- 

in 


in  thys  Pro- 
fofithn. 


c  inks.  If  therefore  two  circles  cut  the  one  the  other  5  they  haue  not  one  mid  the 
fame  centre :  ^bicb  "Was  required  to  be proued, 

S^Tbe  j.Theoreme,  The  6,  *Propofitionm 

If  mo  circles  touch  the  one  the  other ,  they  haue  not  one  and 
the  fame  centre, 

'  fppofi  that  thefe  two  circles  A  E  Cyj?  OD  E  do  touch  the  one  the  other 
\in  the  poynt  C.  Then  I fay  that  they  haue  not  one  and  the  fame  centre . 
For  if  it  be  pofiible  let  the  point  F  be  centre  Wnto  them  both.  And  (by  the 
firfi  petition)drdw  aline from  F to  Ciand  ( 

drawe  the  line  FEE  at  all  aduentures . 
j. fhdfor  afmuch  as  the  poynt  F  is  the  ten* 
tre  of  the  circle  A  E  C  pther for e(by  the  I 
definition  of  the firH)the  line  F C is  equal! 

Ipnto  the  line  FE.  (Agayne  for  afmuch  as 
the  poynt  F  is  the  centre  of y  circle  C  IDE f 
therefore  (by  the fame  definition)  the  line 
FC  is  equally  nto  the  line  FE .  And  it  is 
proved jhat  the  line  F  C  is  e quail  Imto  the 
line  F  E, "Wherefore  the  line  FE  alfo  is  e * 
quail  lento  the  line  F E}  namely  the  lejfa 
yntoy  greater:~which  is  impofiible.  When 

fore  the  poynt  F  is  not  the  centre  of  both  the  circles  AEC  and  Cfb  E .  In  like 
fort  alfo  may  ive  prone  that  no  other  poynt  is  the  centre  of  both  the fay d  circles. 
If  therefore  two  circles  touch  the  one  the  other:  they  haue  not  one  and  the  fame 
centre:  "Which  "Was  required  to  be  demonft rated. 


In  thys  Propofition  are  two  cafes  :  for  the  circles 
touchyng  the  one  the  other,  may  touch  eyther  within  or 
without » If  they  touch  the  one  the  other  within,  then  is  it 
by  the  former  demonftrationmanifeft,  that  they haue  not 
both  one  and  the  felfe  fame  centre.  It  is  alfo  manifeftif  A 
they  touch  the  one  the  other  without :  for  that  euery  cen~ 
tre  is  in  th  e  middeft  of  hys  circle.  ; 


.  ,  .  1  j  .  .•  •  r  .... 

i-^T'he6.  Theoremc*  The  yfPropofition. 


If  in  the  diameter  of  a  ci 


many  poynt. 


is  not 
the 


ofSucIides  Blementes* *  FoLSd* 

the  centre  of  the  circle,  andfromthat  foynt  he  drawen  ynto 
the  circumference  certaine  right  lines :  the  greate&of thefe 
lines (hall  he  that  line  wherein-  is  the  centre ,  and  tm  wff 
fhad  he  the refidue  of  the fame  line  •And  of  alltbe other  form, 
that  whkh  is  nigher  to  the  line  which pajjeth by  theeqntrejis 
greater  then  that  which  is  more  distant.  Andfromthat  point 
can  fall  within  the  circle  o^eehfdefthekMi  lineoneljtwo 


Vppoflybat  time  be  a  circle  JfB-  C®-:.  mdJe&tfeidiameter  thereof 
ke  JJD .  'And  take  in  it  awyjpoynt  hejides$et \  centre,  of  the.  circle,  ymd 
let  the  fame  he  K  And  let  thexentre  of  the,  circfifi by  tj?e>  i.  *  off  third ) 
he  the poynt  &  jfndfirom  the poynt  E  let ;  theyg  he.  draw w  lento the 
circumference  yfF  C  L D  thefe  right 
the  line  Fyfis  thegreatefl :  and  the 
FID  is  the  left .  And  of  the  other  lines, 
the  line  F  F  is  greater  then  the  line  F 
and  the  line  F  C  is  greater  then  the  line 
F  G .  (Drawe  (by  the firfl  petitionfhefe. 
right  lines  F  Efi E}  and  G  E.  jTnd  fo 
afmuch  as(  by  the  2o»  of the  firfl  fin 
ry  triangle  two  fides  are  greater  then 
the  third  therefore)  lines  E  $  and  E  F 
are  greater  then  the  refidue  ,  namely 
then, the  line  F  F.  Fut  the  line  AE  is  e* 


qihlllmto  the  line  FE(bythe  i$*defi* 

nitionof 'the firftfiWherefore the  lines  BE  and  E  Fare -equal!  lento  the  line 

*AF.  Wherefore  the  line  ^AF  is  greater  then  then  the  line  FF,  Agayne  for 
afmuch  as  the  line  F  E  is  equal!  pntoC  E  (by  the  15.  definition  of  thefirft)and 
the  line  FE  is  common  Imto  them  both ^therefore  thefe  two  lines  FEand  EF 
are  equalhmto  thefe  two  C  E  and  EF.  Fut  the  angle  FEF  is  greater  then  the 
angle  CEF.  Wherefore  ( by  the  2  4..ofthe firfl)  the  bafe  F  F  is  greater  then 
the  bafe  CF :  and  by  the  fame  reafon  the  line  CF  is  greater  then  the  line  FG. 
Agayne for  afmuch  as  the  lines  G  F and  F  E  are  greater  then  the  line  EG  (by 
the  zo.of the firfl ) .  Fut(  hy  the  15. definition  of  the firftfihe  line  E  G  is  equal! 
Ipnto  the  line  E  D :  Wherefore  the  lines  G  F  and  F  E  are  greater  then  the  line 
E  Djtake  away  E  Fytybicb  is  comon  to  the  bothptoherforey  refidue  G  F  isgrea * 
ter  then  the  refidue  FD ;  Wherefore  the  line  F  A  is  the greatest }and  the  line 
FD  is  the  lefl3and  the  line  FF  is  greater  then  the  line  FC,  and  the  lineFC 


ConflmBim* 


The  firfl  p*rt 
of  this  Props* 
fitlOHo 

Demonflrt* 

thfSo 


Second  part. 


Third  part .  h  greater  then  the  line  F  G .  Flow  alfo  I fay  that  from  the  poynt  F  there  can  he 

drawen  onely  two  equall  right  lines  into  the  circle  jfBCD  onechefide  of  the 
leaf  line  gamely  F  ID .  For  (by  the  2% .  of the firfl)  lopon  the  right  line geuen. 
B  F  and  to  the  poynt  in  it, namely  E,  make  "onto  the  angle  G  E  F  an  equall  an * 
gle  FE  H:  and(by  the firft  petition)draw  a  line  from  F  to  H.  Flow forafmuch 
as  (by  the  15.  definition  of the  firft )  the  line  E  G  is  equall  Tmto  the  line  E  H, 
and:  the  line  EF  is  common  Wnto  them  both,  therefore  thefe  two  lines  GE  and 
EF  areequalhmto  the fe  two  lines  HE  and  E  F,  and  (by  conftruclion)  the 
angle  G  E  F  is  equall  into  the  angle  HEF .  Wherefore(by  the  4. of y  firft} 
This  demon-  thebafe  FG  is  equall  lonto  the  baj'e  F  H.  I fay  moreouer  that  from  the  poynt 
firmed  by  an  jp  Qm  ye  qrawen  mt0  a'rc/e  no  other  right  line  equall  lento  the  line  F  G  „  For 
dmg  to  an  im-  if  it  pofiible  let  the  line  F  KJbe  equall  lonto  the  UneFG .  And for  afmuch  as  F  If 
geftbiiie.  }s  equall  lonto  F  G .  (But  the  line  F  H  is  equall Imto  the  line  F  G,  therefore  the 
line  F  If  is  equall  lonto  the  line  F H.  Wherfore  the  line  which  is  nigherto  the 
line  which pajfeth  by  the  centre  is  equall  to  that  which  is  farther  of  which  He 
haw  before proued  to  he  impofiible . 
iAn  other  de-  Or  els  it  may  thus  be  demonftrated. 

ofT&tter  bDraw  (  by  the firft  petition )  a  line from 

part  of  the  E  to  If:  and for  afmuch  as(byy  15*  de* 

leadings  find™  °fj firft)y  He  GE  is  equall  lonto 

to  an  mpofii-  J  line  &  K.> dn^ doe  fine  F  Eis  common 
frilitit.  to  them  botb}and  the  bafe  G  F  is  equaU 
lonto  the  bafe  Elf,  therefore  (by  the  S. 
of the firft)  the  angle  GEE  is  equaU  to 
the  angle  IfEF.  But  the  angle  GEF 
is  equall  to  the  angle  HEF .  Where* 
fore  (by  the  firft  common  fentence)  the 
angle  HEF  is  equall  to  the  angle  IfE  F  the  lejfe  lonto  the  greater :  which  h 
impofiible .  Wherefore  from  the  poynt  F  there  can  he  drawen  into  the  circle  m 
other  right  line  equall  lonto  the  line  G  F.  Wherefore  hut  one  onely .  If  therefore 
in  the  diameter  of a  circle  be  taken  any  poynt, which  is  not  the  centre  of the  cir * 
demand from  that  poynt  he  drawen  Imto  the  circumference  certaine  right  lines : 
the greateft  of  thofe  right  lines  Jhall  be  that  wherein  is  the  centre :  and  the  leaft 
jhallbetherefidue .  jindof  all  the  other  lines , that  which  is  nigherto  the  line 
which  pajfeth  by  the  centre  is  greater  then  that  which  is  more  dift ant.  And  from 
that  poynt  can  fall  within  the  circle  on  ech fide  of the  leaft  line  onely  two  equaU 
right  lines :  which  Was  required  to  be  proued. \ 


*4.  CoroHary* 


fj  H  Corollary, 

Hereby  ir  is  manifeft,  that  two  right  lines  being  drawen  fro  any  one  poynt  of 
the  diameteryJhe  one  of  one  fide,and  the  other  of  the  other  fide,if with  the  diame¬ 
ter  they  make  equall  anglcs,thc  fayd  two  right  lines  are  equal!.  As  in  thys  place 
are  the  two  lines  F  G  and  FH. 

fTh 


ofSuclides  Elementes .  FoL  3  Jo 

y&T'he  7 .  FTheoremc.  7  he  S.Tropoftion . 

Ifwithouta  circle  be  takpn  any poynt, and  from  thatpOynt  be 
drawn  into  the  circle  vnto  the  circumference  certayne  right 
lines  ,of which  let  one  be  drawn  by  the  centre  and  let  the  re fl 
be  drawn  at  M  adventures :  thegreatejl  ofthofe  lines  which 
fall  in  the  concauitie  or  hollownes  of  the  circumference  of  the 
circled s  that  which  paffeth  by  the  centre :  and  of  all  the  other 
lines  that  line  which  is  nigher  to  the  line  which  pajfeth  by  the 
centre  is  greater  then  that  which  is  more  diflantfBut  ofthofe 
right  lines  which  end  in  the  conuexe  part  of  the  circumfe¬ 
rence ,th  at  is  the  leaf  which  is  drawen  from  thepoynt  to  the 
diameter:  and  of  the  other  lines  that  which  is  nigher  to  the 
leafisalwaies  leffe  then  thatwhich  is  more  dif ant.  And  from 
th at poynt  can  be  drawen  vnto  the  circumference  on  ech fide 
of  the  leaf  onely  two  equall  right  lines . 


9 


FfMd  Fppofe y  t&e  circle geuen  be  A’B Cx 
without?  circle  AS  C,  take  the 
■  ^d^point  ID :  and froy fame  point  draw 
certain  right  lines  intoy  circle  ynto  the  cir* 
cumference ,  zs  let  the  be  D  A ,D  E,D F, 

<Zsr  D  C:zsr  lety  line  D  A  paffe  byy  centre. 

T  hen  I  fay,  ofy  right  lines  H vhich  fall  in  the 
concauitie  ofy  circumference  A  E  FCy  is, 
loithiny  circle y greatef  isy  Svhich  paffeth 
by  y  centre, that  is,  D  [A.  And  of  thofe  lines 
vhich  fall  ypony  conuex part  ofy  circumfe* 
rencefj)  lef  is  y  iohich  is  drawen  froy  point 
D  lontoy  end  ofy  diameter  yl  G.  And  of  the 
right  lines fallmglbin  the  circumferece ,  the 
line  D  E  is  greater  theny  line  D  E,Zy  the 
line  D  F  is  greater  theny  line  D  C.  jind  of 
the  right  lines  t>bicb  end  iny  conuex  part  of 
the  circumference  J  is,  -without}  circle  ,that 
ybich  is  nigher  yntoD  Gy  lef, is  alwayes  leffe  theny  1* hich  is  more  dipt, that 
is, the  line  D  J\  is  leffe  then  the  line  D  E,and  the  line  D  E  is  leffe  then  the  line 
D  H.  T  ake  (  by  thefirfi  of the  third)thc centre  of the  circle  J.B  C,  and  let  the 

Bb.j.  farm 


The  first  part 
of  this  Propo - 
fimu. 


Sfcsnd  part. 


Third  part. 


fame  be  M  :  and ( by  the  firji petition)  drawe  thefe  right  lines'  ME,MT?.JMC, 
MB,  ML  ?and  M If.  find for  afmuch  as  (by  the  15  .definition  of  the  ftrft) 
the  line  MM  is  equall Icyito  the  line  E  M, put  the  line  MD  common  to  them 
both .  Wherefore  the  line  AID  is  eqnall  Imto  the  lines  EM  and  MD .  ‘But 
the  lines  E  M  and  M  D  are  ( by  the  2  o.  of  the ftrft) greater  then  the  line  E  D: 
Wherefore  the  line  MDaljo  is  greater  then  the  line  E  D .  Agaynefor  afmuch 
as  (by  the  definition  of  the  ftrft)  the  line  ME  is  eqnall  lento  the  line  ME, 
put  the  line  MD  common  to  them  both '.Wherefore  the  lines  EM  andMD 
are  eqnall  to  the  lines  FMandM  D,and  the  angle  E  MD  is  greater  then  the 
angle/PM  D :  Wherefore  (by  the  19*  of the  firji)  the  bafie  E  D  is  greater  then 
the  bafie  FD  .In  like jort  alfio  may  Ice  prone  that  the  line  ED  is  greater  then 
the  line  C  D .  Wherefore  the  line  D  A  is  thegreateft,and  the  line  D  E  is  grea¬ 
ter  then  the  line  D  Fj  and  the  line  D  F  isg, 

jfnd  for  afmuch  as  (  hy  the  20.  of  the 
firji)  the  lines  MJf  and  lfjD  are  greater 
then  the  line  M  D .  ‘But  (by  the  i^.defini* 
tion  of the  firji)  the  line  M  G  is  equal!  J>n* 
to  the  line  M  If  .Wherefore  the  ref  due 
KJD  is greater  theny  refidue  G  D.VVher* 
fore  the  line  G  D  is  lefife  then  the  line  KJD. 

.Andjor  afmuch  as  from  the  endes  of one  of 
the fides  of  the  triangle  MED ,  namely, 

M  D  are  drawen  two  right  lines  M  If  and 
KJD  meeting  within  the  triangle ,ther fore 
(hy  the  21.  of  the  firji)  the  lines  M  If  and 
KJD  are  lefife  then  the  lines  ML  if  ED, 
of  mhich  the  line  M If  is  equal!  Imto  the 
line  M  E  .  Wherefore  the  refidue  D  If 
is  lefife  then  therejidue  D  E  .  In  like  jort 
alfio  may  me proue  that  the  line  D  E  is  lefife 
then  the  line  D  El .  Wherefore  the  line 
D  G  is  the  left,  and  the  line  D  If  is  lefife  then  the  line  D  L,  and  the  line  D  E 
is  lefife  then  the  line  DH. 

Now  alfio  I Jay  that  from  the  pqyntlp  can  be  drawen  Tnto  the  circumference 
6n  eche  fide  of  DG  the  leaft  onely  two  equal!  right  lines .  Vpon  the  right  line 
MD,  andlmto  the  poynt  in  it  M make  (by  the  2%:  of the  fir  ft)  lento  the  an* 
gle  KJMD  an  equal!  angle  DMfi .  And  (by  the  ftrft petition)  drawe  a  line 
from  D  toB.  Jbnd  for  afmuch  as  (  by  the  l^definition  ofthefirU  )  the  lint 
M  B  is  equall  lento  the  line  M- If  put  the  line  MD  common  to  the  both,mher* 
foye  thefe  two  lines  M  If  and  M D  are  equall  to  thej'e  two  lines  B  M andMD 
the  one  to  the  other,  and  the  angle  LfM  D  is  (  hy  the  2\,  of  the ftrft)  equal!  to 
the  angle  B  MD:  IWherefore  (  by  the  4.*ofthefirU  )the  baft  D  If  is  equall 


FolM. 


.  to  the  hafe  D  3 . 

ISLow  I  fay  that  from  the poynt  D  on 
that  fide  that  the  line  ID  3  is,  can  not  he 
drawen  Imto  the  circumference  any  other 
line  hefides  D  3  equall  lento  the  right  line 
DAfePor  fit  he  pofitble  let  there  he  drawen 
an  other  line  hefides  D  tB, and  let  the  fame 
be  D  N.  y(nd for  afinuch  as  the  line  D  If 
is  equall Imto  the  line D  N.  But  lento  the 
line  D  Jf  is  equall  the  line  D3.Therfore 
( hy  the  fir  ft  common  fentence)the  line  D  3 
is  equall  lento  the  line  DK.  Wherefore 
that  ‘ft Inch  is  Higher  lento  ID  G  the  least, is 
equall  toy  "Which  is  moredifiant:  Which 
-We  haue  before proued  to  he  impofiihle. 

Or  it  may  thus  he  demonstrated »  Draw 
(hy  the  firfl  petition  )  a  line  from  M  to  'N.- 
And  for  afinuch  as  (hy  the  15*  definition  of 
thefirfi)the  line  IfiM  is  equall  lento  the 
line  MN,and  the  line  M D  is  common  to  them  both .  And  the  hafe  JfD  is  c* 
quail  to  the  hafe  D  IS!  (hyfuppofition  )  therefore(hy  the  ft  *  ofthefirfi )  the  an* 
gle  KfKFD  is  equal!  to  the  angle  DMpZ  ,3ut  the  angle  KfiAD  is  equall  to 
the  angle  3MD.  JVberfore  the  angle  3  M  D  is  equallto  the  angle  N  M W, 
the  lejje  lento  the  greater:  -which  is  impofiihle :  Wherefore  from  the  poynt  D  can 
not  he  drawen  lento  the  circumference  A3  C  on  echefide  of  DO  the  left,  more 
then  two  equall  right  lines .  If  there  fore  without  a  circle  be  taken  any  poynt  and 
from  that  poynt  he  drawen  into  the  circle  lento  the  circumference  certaine  right 
lines, of -which' let  one  be  drawen  by  the  centre , and  let  the  resihe  drawen  at  all 
aduenturc-s : the greatefi  of  thofe  right  lines -which fall  my  concauitie  or  hollow * 
nes  of  the  circumference  of  the  circle  is  thatyhichpafiethby  the  centre .  Jhdof 
all  the  Other  lines , that  line  -which  is  nigber  to  the  line  -which  pafieth  hy  the  cen ! 
ire, is  greater  then  that  -which  is  more  dittant.3ut  of thofe  light  lines  ' which  end 
in  the  conuexe part  of  the  circumference ,  that  !me  is  the  Mt  -which  is  drawen 
from  the  poynt  to  the  dimetient :  and  of  the  other  lines  that  -which  is  nighef'tb 
the  leaf  is dlwayes  leffe  then  thatyhicl:  is:  moredifiant .  ^And from  that  poynt 
Can  be  drawen  Imto  the  circumference  on  echfidtof  thelefi  only  two  equall  right 
lines :  -which  -Was  required  to  hefrbuedl  y  :  r 

A"., ’  \  o*.  KOiVa'.u  ;  A/  'A  \ck.v;  . \  Ax~’1  '  v; 


Thys  Proportion  is  called  commonly  in  oldbookes  amongeft  the  barbarous, 
C/udaPaupms,,  that is,  tfre  Peacocks  taile, 

\j  1  .y  i,.  -  .  i,  \  J  -  >  'i  v  i  i  *  A  v  ■  *\  ■‘•a  \  •  iJf  ■  ;  -■  /  '  ’  '  * 


IS  vv. 


wjtmm. 

i  ■  "  -  TA  yyyr  -  <y  ,  t  ■ -y , 

Hereby  it  is  mani£efl>that  the  right  lineSjWhich  being  drawen  from  the  poynt 


Bb.ij. 


geuen 


This  is  demo* 

prated  hy  an 
argument  (en¬ 
ding  to  an  &b-' 
’  fnrdity. 


An  other  de- 
monUratton 
of  the  latter 
part  Reading 
alfo  to  an  im- 
pofsibiiity» 


This  Propon¬ 
ents  cemmely 
called  CatuU 
i*ttm>nssa 


Confiructioti. 

TXerwnftrtr 

t(on. 


K' 


gcuen  without  the  circle,  and  fall  within  the  circlc,are  equally  aidant  from  the 
lead, or  from  the  greateft  (which  is  drawen  by  die  cehtre)  are  equal!  the  one  to 
the  other :  but  contrarywyfe  if  they  be  vncqualiy  diftant,whcthcr  they  lightvpon 
rhe  concauc  or  conuexe  circumference  of  the  circlc,they  are  vnequalh  s 

y&The  ZfTheorme.  clhe  yfPropofitm.  f 

If  within  a  circle  he  talpen  a  poynt ,  and  from  that  poynt  he 
drawen  lemto  the  circumference  rrioe  then  two  e quail  right 
linesithepoynttal^en  is  the  centre  of  the  circle. 

I  Eppofe  t hat  the  circle  be  A  IB  C,  and  loithin  It  let  there  be  taken  the 
poynt  ID.  And  from  ID  let  there  be  draieen  lento  the  circumference 
ABC  moe  then  two  equal!  rift  lines ,  that  is,  D  A,D  B,  andD  C 
T  hen  I  fay  that  the  poynt  D  is  the  centre  of the  circle  ABC \  Dram 
( by  the  firfl  petition  )  thefe  right  lines 
JB  and  BC :  and (  by  the  io»  of  the 
frfl  fdeuide  the  into  two  equall partes 
m  the  poyntes  E  and  F:  namely,  the 
line  4  B  in  the  poynt  E,  and  the  line 
B.C  in  the  poynt  F ,  And  drqpy  lines  y 
E  D  apd  F  Dy  and  (by  the  fccond pe* 
tition)extendthe  lines  E  D  and  FD 
(}U  eche  Jide  to  the  poyntes  Jf,  G,and 
fif .  A  nd for  afmnch  as  the  line  AE 
as.  equal!  lento,  the  line  E  B,and  the  line 
E  D  is  common  to  them  both, ,  there * 
fore  thefe  two  fdes  A  E  and  %D are  equal!  tynto  thefe  two  J 
B  D :  and.(  byfuppoftion )  the  bafe  D  jf  is  equal!  to  the  bafe  D  B .  Wherfon 
(by  the  8  .  of  thefirft)  the  angle  A  ED  is  equal!  to  the  angle  B  ED  Wherfore 
eyther  of  thefe  anglepji  E  D  aiid,  B  E  D  is  4  fight  angle.  Wherefore  theftiig 
G  Jfjleuidethy  line  A  B  into  two  equall  partes  andmaketh  right  angles.  And 
for  .afmuch  as,  if  in  'a  circle aright  lipe  denude  an  other  right  line  into  two  equaJl 
parte?  in  fuel  fort  Eqtitniafjyqfrftafes^f y  line  that  dcuideth  isth 
cenjpe  ft  he  circlfby  the  Coirolfy  of  ihefirfi  oft  he  third) ,  T  her  fore  ( by  the 
fame  Corr  diary)  _m  the  lino  G 

fame  reafon)maysxe prone  that  in  y  line  HE  is  the. Centre  of  the  circle  A  BC, 
and  the  right  lines  G  Jf,  and  FI  E  bane  no  other  poynt  common  to  them  (oth 
hejides  the poynt  D:  Whereforedhe  poyfit  D  is  the  centre  of  the  circle  A  BC. 
Jf  therefore  within  a  circle  be  taken  a  poynt , and  from  that  point  bedfdweii 
th  e  circumference  more  then  two  equal)  right  hjties,  the  poynt  taken  is  the  centre 
of  the  circle:  Svhicblvas reauiredto.be proued. 

'  ■  -  '  ••  tv  •-••••  4  ■-  *  - ,  *‘J  ‘  '  /’  *•  *  "  **J®- 

'  fjn 

‘httni  -x-atrk  1  7/ 


■ 

c 


A  a. 

.  3V.\  0. 

.  vh 


E:'.  S 


I 


ides 


.rt.oa 


to  an  impofii 
bilitk*  * 


FoL  8p. 

ft  An  other  demonstration. 

;  ■  S  ■  G\  ifcjk  fdG  1  .  k  0  k  k  -••• ,  ,v:  r--:dkpff\ 

y‘  'E'ei.thexehe  taken ■^itUn-^^k(^:A.^€^he  fojnt  D.Anifrom.  the poynt  jn  otforje, 
fDlet-thwe^Mdraw'eniyi^rf^&rcmfinxtckfndre  then  two  equally tgbt  Cmesy  mnlimion 
namely }  DA,DB}  and  2)  t.Then  I  fay  thattbe  poynt  ID.  is  the  centre  of  the: 
circle .  For  if  not ,  then  if  it  be pofible 
let  the  point  E  be  the  centre  rand  draw 
a  line  from  Dio  Eyand  extend  D  E  to 
the  poyntes  F  and  G .  Wherefore  the 
line  F  G  is  the  diameter  of  the  circle 
A  EC.  And  for  afmuch  as  in  F  G  the 
diameter  of  the  circle  AFC  is  taken 
a  poynt ?  namely  Dy  1 vhicb  is  not  the 
centre  of  that  circle ,  therefore  ( hy  the 
7 .  of  the  third)  the  lintDG  is ygrea 


ohsVA  TWAO 

teffy  and  the  line  D  C'is  greater  then  (y,  . 

tea  then  the  line  D  A.  ‘But  the  lines  D  CfD  Bfb  A^are  alfo  e quail  ( byfuppofi* 
tion) :  which  is  impofihle .  Wherefore  the  poynt  E  is  not  the  centre  of  the  circle 
ABC.  And  in  likefortmhyWe  prone  that  mother  poynt  befides  D.  Wherefore 

the  poyrtt  D  is  the  centre  of the  circle  ABC:  “Which  Was  required  to  be  troued, 

-x:  ftjfcca jttivvl  5m»s  tHima  J  1  \  \pfipc 

wU'WA 


.  WThe9M 

YUOrtu  .  . .  dn  <&m 


'WV* 


e  io. 

n  ,  ■  •" 

V:>  ::rav. 


option . 


s»'  ■  i 

V‘  V;.,  E'E 


■  k 
■s\  RTOV' 


Id  .  j  k 


A  cm  le  cutteth  not  a  circle  in  moepointes  then  two* 

*Vj  )  tv  i'.V  '-.c  •>  V  . 

^  r  .  i  «.  f\ ,  ’  •  >  t  i .  .  •  1  .’i  ’  iff'1  y  rt,  ■  -  \ 

Or  if  it  le poflible  let  the  circle  A  BC  cut  the  circle  D  B  F.  in  mopohites 
“""qLtben  two  float  is  pi  BfGJ-dyW  R,  And  drawe  lines fro B to  & ?and from  tion  lead in 
M  B  to  rI.Ana(byy  io  .of the firffeuide  either  of  the  lines  BG&B  El 
into  two  equaU partes pi) pointes  '  .......  j 

Jfand  L.  And  by  then*  of  the 
fir  ft) from  the  poynt  Ifraifelop 
yntoy  line  B II  a  perpendicular 
line  KjC>  and  hkewife  from  the 
poynt  E  raife  lop  yntoy  line  B  G 
aperpendi&dqr  line  hlf  and . 
extend  the  line  L  jf  to  the  poynt 
A  y  and  EFIM  to  the  poyntes 
IX  and  E .. And  for  qfitmch  qs:in 
the  circle' k/f  BC  3  the  right  line ' 

A  C  deuideth  the  right  line  B  IE 
rfei 


/srsv.-  (hj 

r 

s 

A'V'  ,-V 

'L 

W  ... 

v\,  • 

\ 

\  ^  /x 

•  •  •  l,  .  1  o 

c 

jrEjfixC 

/  1 
/ 

.  :  ...,\ 


d'd  maketh  'rght  angles  jherfore( by  the  3 .  of  the  third ) 

Bb.itj.  in  the 


An  other de- 
ftion/l  ration 
of  the  fume 
Heading  alfo 
to  an  mpofti- 
kililie. 


m  the  line  AC  is  the  centre  of  the  circle  ABC.  Agaynefor  afmuch  as  in  the 
jeife  fame  circle  A  BC  the  right  line  NXy  that  is  Me  line  ME  deuideth  the 
right  line  B  G  into  two  equall paries  andmaketh  right  angles ytherefore( Wjfthe 
third of  the  third)  in  the  line  NX  is  the  centre  of  the  circle  ABC.  And  it  k 
proued  that  it  is  alfo  in  the  line  , 

X C  ,  'And  thefe  two  right  lines 
A  C  and  N  X  meete  together  in 
no  other poynt  befides  0.  Where* 
fore  the  pqynt  0  is  the  centre  of 
the  circle  ABC .  And  in  like  fort 
may  1 tie  prone  that  the poynt  0  is 
the  centre  of  the  circle  S)EF: 

Wherefore  the  two  circles  ABC 
and  (D  EF  deuiding  the  one  the 
other  haue  one  and  the  fame  cen * 
tre :  yphiclf  hy  the  5  •  of  the  third) 
is  impofible .  A  circle  therfore  cutteth  not  a  circle  in  moepoyntes  then  two^hich 
t>as required  to be  proued. 

'  '  '  '  1  ■  '  '  m 


if  An  other  detnonftration  to  proue  the  fame. 


S  uppofe  that  the  circle  ABC  do  cut  the  circle  (DGF  in  mopoyntes  then 
that  is  jn  Bf}f\and  H,  And  ( hy  the firfl  of  the  third)  take  the  centre  of the 
circle  ABC  and  let  the fame  be  the  poynt  if.  And  draw  thefe  right  lines  IffB? 
JfGy  and  ]\F .  Now  for  afmuch  as 
within  the  circle  !DE  F  is  taken  a  cer* 
taine  poynt  and from  that  poynt  are 

d'rawen  ynto  the  circumference  moe  then 
two  equall  right  lines  y  namely ,  KJBy 
i\G  yand  JfF :  therefore  (by  the  9  ♦  of 
the  third)  If  is  the  centre  of  the  circle 
ADEF .  And,  the  poynt  If  is  the  centre 
of  the  cirble  ABC.  wherefore  two  cir* 
des  cutting  the  one  the  other  haue  one 
and  the  fame  centre :  "tyhich( by  the  £>  ♦  of 
the  third)  is  impofiible.X circle  therfore 
cutteth  not  a  circle  in  moe  pointer  then  two :  ‘tohich  itas  required  to  hi  demom 
ft  rated. 


5 &T he  IQ.  Theoreme .  The  if  fProftofition* 

If  two  circles  touch  the  one  the  other  inwardly  ,  their  centres 

being 


r-x 


ofEuclides  Elementes.  Fo/.po* 

inggeuen ;  a  right  line  ioyning  together  their  centres  and 
1  will fall  upon  the  touch  of  the  circles. 


Vppofe  thM  jf&C,  and  ASJ-E  do  touch  the  one  the 

other  in  the  poynt  A.  And  (by  the firjl  of the  third)  take  the  centre  of 
^  the  circle  A  £>  C^andlet  the fame  he  F:  and  likpivifey  centre  oft  he  circle 
4fD  Ejand  let  tf  e fame  be  Or-.  Then  l fay  that  aright  line  dnxwen  from  F  to  G 
and.behgprodnccifAill fall  yponthe  poynt  j[.  For  if  not 3  then  fit  he  pofiible 
let  it  fall  as  the  line  F  Q  ID  Fd doth  .  And  dmwthefe  right  lines  A  F}i?  A  G. 
ror  afmnch as  the  lines  A  G  and 


QF  are  (by.the  2:o,  of  thefrsfgrea* 
ter  then  the  line  F  A?  that  is,  then  the 
line  F  H fake  aipay  the  line  GF yehich 
is  common  to  them  both.  Wherefore  the 
ref  due  A  G  is  greater  then  the  ref  due 
G  EL .  Tut  the  dine  T)  G  is  equall  t>nto 
the  line  G A  (by  the  W*  definition  of 
the  fir f),  Wherefore  the  line  G  dp is 
greater  theny  line  G  H:  the  leffe  then 
thegreaterilohich  is  impofiible.  Wher * 
fore  a  right  line  drawen  from  the  poynt 

ftp  fie  poynt  G  and produced fal\eth  not  befides  the  poynt  J,  loUch  is y  point 
wal  rxsl„vhfor'e  itftllethypon  the  touch  ,  If  therefore  two  emits. imch 


quire d  to  beproued. 

'  \\  •  .•*:  r  •  ;  •  q  ’  •  <v  ;  Vv;r;  v.“i  v  g-. :  \  '-j  - ;} 

An  other  demonftration  to  prone  the  (lime. 


Tut  now  let  it fall  as  GF  C  fallethfnd  extendy  line  G  F  Cto  the  poynt  H: 
and  dmwe  thefi  right  lines  J  G  and  A  F.  Arid for  afinuch  as  the  lines  JG  and 
O  F  are  (  by  the  20»  of  the fir fl) greater  then  the  line  A  F.  Tut  the  line  A  F  is 
equall y>nto the  line  CF,  that  is,  ynto  the  line  FH.  Take  away  the  line  FG 
common  to  them  both .  Wherforethe  refidue  A  G  is  greater  then y  refidue  GH 
that  is  fire  line  GJD  is  greater  then  the  line  G  H:the  leffe  greater  then y grea *. 
ter :  lohich  is  impdfiiblel '  1  ft 


Which  thing  may  alfo  be  proued  by  the  7.Propofitibn  of  this  booIce.For  for  afmuch 
as  the  line  H  C  is  the  diameter  of  tnc  circle  A  B  C ,8c  in  it  is  taken  a  poynt  which  is  not 
the  centre, nimely,the  poynt  G,  therefore  the  line  G  A  is  greater  then  the  line  G  H  by 
the  fayd  y.Propofmon .  But  the  line  G  D  is  equall  to  the  line  G  A  (by  the  definition  of 
a  circle)  .Wherefore  the  line  G  D  is  greater  then  the  line  G  H,namely}the  part  greater 

then  the  whole'-:  which  is  irtipoffible.  —  -  ■  -  -  0 


Conjlruftion, 


Demonftra- 
tion  leading 

jiUlitti. 


An  other  de* 
man  ft  ration 
of the  fame 
leading  alfo 
to  an  mpeffla 

Wti'fi 


J  he  fame  4* 
gaine  demon* 
jhated  by  an 
Argument  lea « 
ding  to  an  ab* 
fur d it  it  is. 


XU 


o 


•  .  -.0 


rr  *  r  ''  7  ■  /  -  t  ■  '/  '  V&J  > -v':  | 

If  wo  circles  touch  the  one  the  other  ouwardly^a  righflme 

,  '  .  .  ri'G.W  .K  Av(pO;  A\  kW 

'  \»  V«  C  .  .  '  ‘  .  ^  '  .'y  I  yTi"’  uV 

ig||  VppofewlMjT^fe  tw&Sh'M  A*B  C  and  ft:lDE  do  touch  flkone  the  o* 


And  (by  the  third  of  the  third')  take  the 


Umonjlnh 

tie  leading  to 
or/  tmpojfibi- 
titic . 


the  right  line  F  C  ID  G  doth.  And 
draw  thefe  right  lines  A Fcr  AG. 

„ And  for  cijmuchas  the  poynt  F  is 
the  centre  of  y  circle  JFB  Cyber* 
free  the  line  FA  is  equall  Tmto 
the  line  FC .  Again  e  for  afmuch 
as  the  poynt  G  is  the  centre  of  the/ 
circle  A  iD  E. ,  therefore  the  line 
G  A  is  equall  to  the  line  G  ID.  And 

And  it  is proued  that  the  line  F  A  is  equall  to  the  Me  VC  .  Wherefore  the  Tines 
FAmdAG  dr/equall  lento  'the -lines' F Cand'G"T>f  Wherefore  the  Ipbole  line 


poynt  G  jhall  paffe  by  the  poynt  of  the  touch  yiamefy  }hy  the  poynt  A .  If  there «*' 
fore  two  circles  touch  the  one  the  other  outwardly ,  a  right  line  drawen  by  their 
centres  Jhall  paffe  by  the  touch :  "ft?  Inch  leas  required  to  be  demon f  rated, 

•  w  ^  t  *  .  •  -  ,L  .  _  , 

ff An  other  demorf  ration  after  Telitarius. 


An  other  de - 
monfiration 
after  Feliu- 
pj  its  leading 
at  Jo  to  an  ah- 
jurditie , 


V-\ 


\  \  iy 


Ka 


r-y 


Suppofe  that  the  two  circles  ^4  B  C  and  T>EF  do  touch  the  one  the  other  out* 
watdly  in  the  poynt  fsfi  And  let  G  be  the  centre  of  the  circle  tABC :  From  wbic-h 
p.oynr  produce  by  the  touch  of  the  circles  the  line  G’^tothc  poynt  F  of  the  circum¬ 
ference  DEE .  Which  for  afmuch  as  ir.paftetjinot  by  the  centre  of  the  circle  2>  Efr 
(as  the  aduerfary  affirmeth  )  draw  , 
from  the  fame  centre  G  an  other  , 
right  line  G  A',  which  - if  itbepoffi*  ) 
ble  let  paffe  by  the  centre  of  the  cir¬ 
cle  D  E  F ,  namely,  by  the  poynt  Hi 
Cutting  the  circumference  tABC  F 
in  the  poynt  B ,  &  the  circuference 
DEE  in  the  poynt  D  y&let  theop- 
pofite  poynt  therof  be  in  the  'point 
X.  And  for  afmuch  as  fro  the  poynt 
Cj  taken  without  the  circle  D  E  F  is 
drawen  the  line  G  K  paffing  by  the  centre  M  and  fro  the  fame  poynt  is’drawen  alfo  an 

©thee. 


/ 

i  ' 

AaT 

0, 


mentes , 


FoL9  T, 


other  line  not  patting  by  the  centre,namely, the  line  G  F .  Therefore  (by  the  § .  of  thvs 
bookc)  the  outward  part  G'T>  of  the  lipe  G  K  (hall  bejeffethen  the  outward  part  G  A 
oftheline  G  F.But  the  line  G  Ais  equal!  to  the  line  G  B .Wherfore  the  line  G  D  islefie 
then  the  line  O' ^namely  ,the  whole  lefll*  then  the  part ;  which  is  abfurde. 

y&The  12.  T bcoreme.  The  13.  Tropofition* 

A  circle  can  not  touch  another  circle  in  mce  foyntes  then 
one  whether  they  touch  within  or  without. 

Or  if  it  be pofsible  Jet  the  circle  ABC  D  touchy  circle  EB  FD 
firfi  inwardly  in  moc poyntes  then  one  ghat  is  fin  D  an  dB. Trike 
I  (by  the  firft  of  the  third )the  centre  of  the  circle  A(B  C  Dyand  let 
1  the  fame  bey  point  G;.and  hkewjfej  centre  of  the  circle  E'BFtD, 
and  let  the  fame  hey  poynt  H.  Wherefore( by  the  11.  of  the  fame ) 
a  right  lh\e  drawen  from  {he  poynt  G  to  the* poynt  IT and, produced pttillf  ill  yp- 
on  the  poyntes  B  and  D  :■  let  it  jo fall  as  the  line  BG  HD  doth.  And for  ufmuch 
as  the  poynt  G  is  the  centre  of the  circle  ABC  Dgherejore  (by  the  15*  definiti* 
on  of  the  f  rfi)  the  line  B  G  is  e quail  to 
the  line  ID  G ,  Wherfore  the  line  B  G  is 
greater  then  then  the  line  H  D:  Where 
fore  the  line  B  FI  is  much  greater  then, 
the  line  FID  ,  Agrime  for  a f  nuch  as 
the  poynt  FI  is  the  centre  of  the  circle 
EBFD ^therefore  (by  the  fame  defini* 
trim)  the  line  BH  is  equal!  to  the  line 
HD :  audit  is  proued  that  it  is  much 
greater  then  it:  yphichis  imjxfiible..  H 
circle  thewfwO  can  not  touch  a  cirelriin 


Of circlet 
Dehicb  touch 
the  one  the 
other  mp$ri~ 

& 


r  a  circle  Pouchyth  a  circle  iri-nioe 
poyntes  then  one. For  if  it  he  pofrible  jet 
thy  circle  A  C  touchy  circle  A  BCD 

outwardly  in  moc  poyntes  theme ,thafz- 
is  jn  A  andC :  And  (by  the firft petition)Arriw  aline  from  the  poynt  \AtqtFT 
poynt C.  Now  for  afmuch  as  in}  circumference  cfeith&(ftkf  ci fries  HBC  ' 

and  A  C  Ff^are  taken  two  poyntes- at  all  adiknturesgAmefryJ^dTjhe^rr 
(by  the fecofijfrtFe third)  a  righstfinetoynihg  together  tfoW p(yhtesjhal(fii- 
^ithin  both  the  circles .  Bui  itfalleth 1 mtlm  the  circle  AB  t.Dj?  Without  the 
circle  AC  KJ  nvhichis  abfurde  ./  Wherefore  a  circle  fhall not  touch  a  circle,  out* 
wardly  in  nioepointes  then  one Arid it  is prouedy  neither  alfbinwardlyyyFereF 
yore  a  circle  orimw&tp&Ghph other  circle  in  moc: poyntes  then  oney  Whether  they 


t)f  arch) 
H>hich  touch 
the  one  the 
other  out- 
fpkrdly. 


touch 


tvucbmthm  or  without:  lohkh  Tx>as  required  to  be  demonflrated. 


\An  other  de¬ 
monstration 
after  Pehta- 
rtas  zr  Fluf - 
fates, of  circles 
Jebicb  toocb 
the  one  the 
ether  out- 
‘&4rdiy» 


Of  circles 
which  tooch 
the onr  the 
other  in¬ 
wardly. 


f  ^Another  demonstration  after 4Pelitarius  and  Fluff ates. 

Suppofe  that  there  be  two  circles  AB  G  and  AD  G,  which  if  it  be  poffible,iettouc& 
the  one  the  other  outwardly  in  moepoyntes  then  one,  namely,  in  A  and  G.  Let  the 
centre  of  the  circle  AB  G  be  the  poynt  I,  and  let  the  centreof  the  circle  ADG  be  the 
poynt  K.  And  draw  a  right  line  from  the  poynt  I  to  the  poynt  K,  which  (  by  the  12. 
of  thys  booke  )  fhall  pafle  both  by  the 
poynt  A  and  by  the  poynt  G:  which  is 
not  poffible  :  for  then  two  right  lines  . 

Ihould  include  a  fuperfiejes, contrary  to. 
the  laft  common  fentence.  Itmayalfobc 
thus  demonftrated.Draw  a  line  from  the 
centre- 1  to  the  centre  K,  which  fhalipalfe 
by  one  of  the  touches,  as  for  example  by 
the poynt  A.  And  draw  thefe  rigbtlines 
G  K  and  G  I,  and/p  (bail  be  made  a  tri-  . 

'•  <nglc,  Vhofetwo  fides  G  K  and  G I  Hull  not  be 
contrary  to  the  2o.dfth,efirft. 

*  Bi;  brow  if  it  be  poffible, let  the  forefayd  circle  ADG  touch  the  circle  A  B  C  inward- 
lyin’ woe.  poyntesrthen  bne,namelv,in  the  pointes  A  and  G :  and  let  the  centre  ©f  the 
circle  ABG  be  the  poynt  I  ,  as  before  :  and  let 
the  centre  of  the  circle  ADG  be  the  poynt  K,  as 
alfo  before.  And  extend  a  line  from  the  poynt  I 
to  the  poynt  K,  which  lhall  fall  vpon  the  touch 
(by  the  1 1  .of  thys  booke) .  Draw  alfo  thefe  line* 

KG, and  IG  .  Andforafmucha^  theline  KG  is 
cqualltothelineKA(by  the  1 5. definition  of the 
firfl)  adde  the  line  KI  common  to  them  both. 

Wherefore  the  whole  line  AI  isequalltothetwo 
lines  KGandKI:  but  vnto  the  line  A I  is  eq  ual! 
theline  I G  (by  the  definition  pf  a  circle)*Wber~,v 
fore  in  the  triangle  IKG  the  fide  IG  is  not  lefic  ' 
then  the  two  fides  IK  and  KG;  which  is  con¬ 
trary  to  the  20.  of  the  firft. 


The  i j.  Theoreme.  *The  14..  Tropofition. 


vimA  \0 

-Wtw'Alet 

The  fir  ft  part 
of  this  Theo¬ 
rems. 


Conflruftion. 


In  a  circle, fe  quail  right  lines ^eequ^diflatitfiim 
tre.  And  lines  equally  diTtantfrom  tie  centre, 
one  to  the  other.  w.iddv  v' - 

■ .  .  /  \  cb  k  ' 

v  >  J  a 

rffoj'e  that  there  he  a.  circle 
{  C there, he  in  it 
Tvy  drawM :  thefe  equall  right,  fines 
4  ®  ®  ^  hen  lj^that-th^,4t’e[ 

equally  dijiant,  from,  the  centre,  X <tke(hy 


ckf 4$  CdDyOfifUt  thefd^h?yfoyntf[  ~  ^  . 
pint  tk.line^4S<^eaX' ;  ,  T  ,t 


i  e, 


c.  -  "V 


X'l  • 
j  ./ 


|  s\  *.r  W  «-•* 

•  ;  .W  'wi  -Vv 


perfmdicxkr 


ofEuclides  Elemcntcs. 


Fol.p 


Am  ft 


perpendicular  tines  EF  and  EG.  And  ( by  thefuft  petition)  draw  thefe  right 
tines  A  E  and  C  E .  Flow  for  afmuch  asa  ccrtaine  right  tine  E  F  drawen  by  the 
centre  cuttetha  certaine  other  right  tine  A  B  not  drawen  by  the  centre ,  in  fuck 
fort  that  it  make  th  right  angles  ^therefore  (by  the  third  of the  third )  it  deunleth 
it  into  two  equall paries .  Wherefore  the  line  A  F  is  equall  to  the  line  FB.Wher * 
fore  the  line  A  B  is  double  to  the  line  AF:  and  by  the fame  reajon  alfo  the  tine 
C T>  is  double  to  the  tine  C  G .  But  the  line  A  B  is  equall  to  the  line  C D.  Wher * 
fore,  the  line  A  F  is  alfo  equall  to  the  tine  C  G .  And  for  of  nuchas  ( by.  the  1 5  *de* 
fnitwn  of  the firf)  the  line  A  E  is  equall  to  the  line  EC,  therefore  thefquare 
of the  line  E  C  is  equall  to  the fquare  of  the  line  A  E .  But  fyhto  the fquare  of the 
tine  A  E,are  equall  (by  the  47-  of thefirft)the fquares  of the  lines  A  F  W  F  E: 
for  the  angle  at  the  poynt  F  is  a  right  angle .  And  (by y  felje  famefto  the  fquare 
of  the  line  E  C  are  equall  the  fquares  of  the  lines  E  G  and  G  C:for  the  angle  at 
the  poynt  G  is  a  right  angle  .  Wherefore  the  fquares  of  the  lines  AF  andFE 
are  equall  to  the  fquares  of  the  lines  CG  and  G  E:  of  lehicb  the  fquare  of  the 
line  AE  is  equall  to  thefquare  of  the  line 
C  Gtfor  the  line  A  Fisrquall  to  the  line 
C  G  .  Wherefore  (  by  the  third  common 
fentence)  thefquare  remay ning ,  namely f 
the fquare  of  the  tine  F  E,  is  equall  to  the 
fquare  remayning ,  namely,  to  thefquare 
of  the  line  EG  :  Wherefore  the  tine  E  F 
is  equall  to  the  line  E  G .  But  right  lines 
are  fay  d  to  be  equally  difant  fromy  cen* 
trepyhen  perpendicular  lines  drawen  fro 
the  centre  to  thofe  lines,  are  equall  (by  the  4*  definition  of the  third).  Wherfore 
'the  lines  A  B  and  C  D  are  equally  difant  from  the  centre. 

But  nowfuppofe  that  the  right  lines  AB  and  CT)  be  equally  difant  from 
the  centre, that  is, let  the  perpendicular  line  EF  be  equall  to  the  perpendicular 
like  EG.  T  hen  1  fay  that  the  line  A.  B -is  equal!  to  the  line  C  T> .  For  the  fame 
order  of conftruRion  remayningppe  may  in  tike  fort  prone  that  the  line  A  B  is 
double  to  the  line  A  F,  dm that  the  line ChD  is double  to  the  line  C  G.  And  for 
afmuch  as  the  line  AE  is  equall  to  the  line  CE,for  they  are  drawen  fromy  cen= 
tre  to  the  circumference ,tber fore  the fquare  of  the  line.  A  E  is  equall  tof fquare 
of  the  tine  CE.  But  (by  the  47.  of tEffir§lfto  thefquare  of  the  line  AE  are 
equall  the fquares  of the  lines  E  F  and  E  A.  And  (by  the felfe fame)  toy  fquare 
of  the  line  C  E  are  equall  the  fquares  of the.  lines  E  G  and  GC .  Wherfore  the 
fquares  of  the  lines  EF  and  FA  are  equall  tothefquares  of the  lines  EGaml 
G  C.  Ofylpch  thefquare  of  the  tine  EG  irequall  to  thefquare  of  the  line  E  i\ 
for  the  tine  E  F  is  equall  to  the  line  E  G,  Wherefore  (by  the  third  common  feu* 
tence)  the fquare  r ern  ayn  ing, namely, the  fquare  of  the  line.  A  E , is  equall  to  the 
fquare  of the  line  C  G .  Wherefore  the  equall '.into  the  tine  C  G.  But 

'  '  the 


T  :  ‘0  ■  r\> 


JjcmonUra- 

tion. 


Demonstra¬ 
tion . 


Dhefecovd 
part  which  i 
she  Conner je 
of  the  rip » 


■An  other  de- 

monjiration 
c ftbefirft 
part  after 
tampans 


■GsnfirutUofi- 


the  line  AB  is  double  to  the  line  A  F}  and  the  line  C  3)  is  double  to  the  lint 
C  G .  Wherefore  the  line  HB  is  equall  to  the  line  C  3)  .  Wherefore  in  a  circle 
equall  right  l hies  are  equally  diftantfrom  the  centre .  And  lines  equally  dijlant 
from  the  centre ^are  equall  the  one  toy  other  :  h vhkh  ~%>as  required  to  be proued 

iff  An  other  demonstration  for  the frf  part  after  Cam  fane. 

Suppofe  that  there  be  a  circle  nA  B  D  C, whole  centre  let  be  the  poynt  £ .  And  draw 
in  it  two  equall  lines  A  B  and  CD.  Then  I  fay  that  they  arc  equally  diftant  from  the 
Centre. Draw  from  the  centre  vnto  the  lines  AB 
and  CD,  thefe  perpendicular  lines  E  F  and  E G. 

And(by  the  2 .  part  of  the!  3 .  of  this  booke  Jthc 
line  A  B  (hall  be  equally  deuided  in  the  poynt  F, 
andtheline  CD  lhall  be  equally  deuided  in  the 
poynt  G.  And  draw  thefe  right  lines  EA,EBt 
EC,  and  ED.  And  for  afmuch  as  in  the  triangle 
mAEB  the  two  fides  tA  B  and  AE  arc  equall 
to  the  two  fides  C  D  and  C  E  of  the  triangle 
C  ED,  &  the  bafe  EB  is  equall  to  the  bafe£Z>, 
therefore  (by  the  8.  of  the  firft)  the  angle  at  the 
point  A  (hall  be  equall  to  the  angle  at  the  point 
C .  And  for  afmuch  as  in  the  triangle  A  E  F  the 
two  fides  AE  and  A  F  are  equall  to  the  two 
fides  CE  andC<j  of  the  triangle  CEG,  and  the 
angl  e.E  A  F  is'equall  to  the  angle  CE  G}  therefore  (by  the  4.  ofthe  firft)thc  bafe  is 

equall  to  the  bafe  E  G :  which  for  afmuch  as  they  arc  perpendicular  lines,  therefore  the 
lines  ABSc  CD  are  equally  diftant  fro  thccentre,by  the  4.  definition  of  this  booke, 

h&The  Theoreme.  The  15.  Trofoftion, 

In  a  circle a  the  greatejl  line  is  the  diameter ,  and  of  all  other 
lines  tha  t  line  which  is  nigber  to  the  centre  is  alwajes greater 
\  then  that  line  which  is  more  diftant* 


Vppofe  that  there  be  a  circle 
ABC  3),  and  let  the  diameter 
thereof  be  the  line  yf3>3and  let 
the  centre  thereof  be  the  poynt  E.  And 
ymto  the  diameter  A3)  let  the  Tine  3  C 
be  nigher  then  the  line  F  G .  Then  If  ay 
that  the  line  A3)  is  the greatejl ;  and 
the  line  B  C  is  greater  then  y  line  FG. 
3)  raw  (by  the  1 2 .  of the  frit) from  the 
centre  E  to  the  lines  B  C  and  FG  per* 
pendicular  lines  EH.  and  Elf  .  And 
for  afmuch  as  the  lineB  C  is nigher  lan* 
to  the  centre  then  the  line  F Gptherfore 


of Bmlides  Elemerites . 


Fol.py 


(by  the  4 .  definition  of  the  third )  the  line  E  If  is  greater  then  the  line  E  H. 
find  (by  the  third  of  the  firft ) put  lento  the  line  E  H  ah  equall  line  E  L.  And 
(by  the  u,  of  the firft) from  the  point  L  rdtfelep  lento  the  line  Ebfia  per  pen* 
dicularline  LM:  and  extend  the  line  L  M  to  the  poynt  7A .  And(bg  the  first 
petition )  drata  thefe  right  lines,  E  M,  E  N,  E  F,  and  E  G .  And  for  afinuch  as 
the  line  E  FI  is  equall  to  the  line  EE ,  therefore  ( by  the  14 *of  the  third,  and 
by  the  4  *  definition  of  the  fame )  the  line  ©  C is  equall  to  the  line  M jSl.  Againe 
for  afinuch  as  the  line  A  E  is  equall  to 
the  line  E M,  and  the  line  .EE)  to  the 
line  EM,  therefore  the  line  A  A)  ise* 
quail  to  the  lines  M  E  and  E  El .  lint 
the  lines  M E  andE  !A  are( by  thezc, 
of the  fir  fi) greater  then  the  line  MIA. 

Wherefore  the  line  yf  A)  is  greater  then 
the  line  MTA .  Mad  for  afinuch  as  thefe 
two  lines  M E  and  E lA  are  equal l  to 
thefe  two  lines  F E  and  EG  (by  the 
15,  definition  of  the  firft  ) for  they  are 
dmwenfrom  the  centre  to  the  circumfe¬ 
rence  ,and  the  angle  ME1A  is  greater 
then  the  angle  F EG ,  therefore( by  the 
24.  of  the  firft)  the  bafe  MIA  is  greater  then  thebafe  EG.  Eut  itisproued 
that  the  line  MIA  is  equall  to  the  line  ©  C  :  Wherefore  the  line  ©  Calfo  isgrea* 
ter  then  the  line  F G .  Wherefore  the  diameter  ME)  is  the  greateft  ,and  the  line 
EC  is  greater  then  the  line  EG .  Wherefore  in  a  circle, the greateft  line  is  the 
diameter  ymd  of  dll  the  other  lines, that  line  "which  is  nigher  toy  centre  is  alwaies 
greater  then  that  line  which  is  more  diftant :  which  Was  required  to  be proued. 

tfMn  other  demonfir ation  after  Campane . 

In  the  circle  eABC  I^whofe  centre  let  be  the  poynt  £,  draw  thefe  lin es^BjAC, 
isf'D,FG3'j.nd  HK,  ofwhich  let  the  line  D  be  the  diameter  ofthecircie.Then  I  lay 
thattheline  AD  is  .the  greateft  ofall  the  lines. 

And  the  other  lines  eche  of  the  one  is  fo  much 
greater  then  ech  of  the  other  ,how  much  nigher 
it  is  vnto  the  centre .  loyne  together  the  endes 
of  all  thefe  lines  with  the  centre ,  by  drawing 
thefd  right  lines  E  B,E  C,E  g,E  K,E  £/,and  E F. 

And  ( by  the  20.  of  the  firft )  the  two  fides  E  F 
and  £  G  of  the  triangle  EE  G,  (hall  be  greater 
then  the  third  fide  F  G  .  Andforafmuch  asthe 
fayd  fides  EF  8c  EG  are  equall  to  the  line  AD 
(by  the  definition  ofa  circle)  therefore  the  line 
tAD  is  greater  then  the  line  ££?.  And  by  the 
fame  reafon  it.  is  greater  then  euery  one  of  the 
reft  of  the  lines, if  they  be  put  to  be  bafes  of  tri¬ 
angles  ;  for  that  euery  two  fides  drawen  fro  the 


Oc.j. 


centre 


Demonjlnt* 

lion. 


An  other  de-* 
monftr  ation 
after  Cam - 

1 fane* 


/ 


centre  .are  equal!  to  the  line  t/C  D .  Which  is 
the  firft part ofr* the  Propofition.  Agayne,for af- 
irmch  as  the  two  (ides  EF  and  EG  of  the  tri¬ 
angle  E  F  Gy  are  eqaall  to  the  two  fides  E  H 
and  EK  of  the  triangle  EH  K,  and  the  angle 
F  EG  is  greater  then  the  angle  H E X,therforc 
(by  the  24*ofthe  firft)  thebafe  FG  is  greater 
then  the  bafe  H  K.  And  by  the  fame  reafon  may 
it  be  proued,  that  the  line  A  C  is  greater  then 
the  line  A  B .  And  fo  is  manifeft  the  whole  Pro- 
$>ofitidn. 

"  r't-  V  *  * ‘  •  w  . 

<: .  >  •'  s 

...  ...  T.  ,  ...  ,  .  . 

Si /The  15. Theorems.  T'he  \6.Tropofition . 

If from  the  end  ofthe  diameter  of  a  circle  be  dr  often  aright 
line  making  right  angles :  it (hall fall  ftithout  the  circle:  and 
betftene  that  right  line  and  the  circumference  can  not  be 
dr  Often  an  other  right  line :  and  the  angle  ofthe femicvcle  is 
grea  ter  then  any  acute  angle  made  of  right  lines ,but  the  0^ 
ther  angle  is  lejje  then  any  acute  angle  made  of  right  lines. 


The  foft  part 
of thts  Theo- 
reme* 

T>emonftra- 
tv'-n  leading 


Appofe  that  there  he  a  circle  AB  C:  whofie  centre  let  he  the  point  D9 
and  let  the  diameter  therofhe  AB.T hen  I  fayy  a  right  line  drawen 
\  from  the poynt  A,  making  with  the  diameter  A I B  right  angles  fh all 
z  fall  without  the  circle .  For  if it  do  not /hen  if  it  he  pofiihle  9  let  it fall 

Within  the.  circle  as  the  line  A  C  doth,  ; 

and  draw  a  line  from  the  point  D  to  the 
point  C.  „ And  for  afmuch  as  (by  the  ifT* 
definition  of  the firfl)  the  line  DA  is 
to  an  ahfurdi-  equal!  to  the  line  D  C  ?for  they  are 
drawen  from  the  centre  to  the  circum * 
ference 3  therefore  the  angle  DAC  is 
equall  to  the  angle  A C  ID.  (But  the  an* 
gle  ID  AC  is  (by  fuppofition)a  right 
angle :  Wherfore  alfo  the  angle  ACID 
is  a  right  angle .  Wherefore  the  angles 
D  AC  and  AC  D /re  equall  to  two 
right  angles :  which  (hy  the  17.  of  the 
firft)  is  impofiihle  .  Wherefore  a  right 

line  drawen  from  the  poynt  A  >  making  With  the  diameter  A  B  right  angles 
frail  not  fall  within y  circle.  In  like fort  alfo  may  We  prone /hat  itfalleth  not  ih 


ofSumctes  Elenientes. 


FoL<)-\. 


the  circumference .  Wherefore  itfalleth  ydthout,as  the  line  A  E  doth . 

I fay  alfo,that  hetwene  the  right  line  A  E,  and  the  circumference  AC  By 
can  not  he  drawen  anothcrrlght  line-.  For  f  it  be pofiiblejet  the  line  AF fo  he 
drawen .  And  (by.  the  1-2.  of the  fir ft) from  the  poynt  ID  draw  tinto  the  line  FA 
a  perpendicular  line  D  G .  And  for  afinuch  as  AG  D  is  a  right  angle  ,  but 
DAG  is  leffe  then  a  right angle , therefore  (by  the  19*0/' the firftfihe fide  A  D 
is  greater  then  the  fide  D  G .  'But  the  line  D  A  is  equall  to  the  line  D  FI,  for 
they  are  drawen  from  the  centre  to  the  circumference .  Wherefore  the  line  D  H 
is  greater  then  the  line  D  G:  namely, the  leffe  greater  then  the  greater  :  ivhich, 
is  impofiible .  wherefore  hetwene  the  right  line  AF.  and  the  circumference 
ACB,  can  not  be  drawen  an  other  right  line. 

1  fay  moreouer ,  that  the  angle  of 
the femicircle  contayyied  "tindery  right 
line  A  B  and  the  circuference  C  HA, 
is  greater  then  any  acute  angle  made  of 
right  lines .  And  the  angle  remay ning 
cotayned  "tindery  circumference  CHA 
and  the  right  line  A  E,is  leffe  then  any 
acute  angle  made  of  right  lines .  For  if 
there  be  any  angle  made  of  right  lines 
greater  then  that  angle  Hnchiscon* 
tayned  "tinder  the -right  line  BA  and 
the  circumference  C  HA,  or  leffe  then 
that  K>])ich  is  contayned  "under  the  cir * 
cumference  CHA  and  the  right  line 
A  ft, then  bettime  the  circumference  CHA  and  the  right  line  A  E, there /hall 
fall  a  right  line ,  "tihich  maketh  the  angle  contayned  Under  the  right  lines, grea* 
ter  then  that  angle  "tihich  is  contayned  "tinder  the  right  line  BA  and  the  cir * 
cumference  C II  A,  and  leffe  then  the  angle  "tihich  is  contayned  Under  the  cir * 
cumference,  C  H  A  and  the  right  line  A  E.  But  there  canfallno  fuch  line,  as 
it  hath  before  bene  proued.  Wherfore  no  acute  angle  contained  tinder  right  lines , 
is  greater  then  the  angle  contayned  tinder  the  right  line  B  A  and  the  circumfe * 
rence  C H  A,  nor  alfolejfe  then  the  angle  contayned  tinder  the  circumference. 
C  H  A  and  the  line  A E. 


Correlarj . 

Hereby  it  is  manifefi  that  a  perpendicular  line  drawen fro  the 
end  of  the  diameter  of 4  circle .  toucheth  the  circle:  and  that  a  right 
line  toucheth  4 circle in  0  ne poynt  onely .  For  it  was  proued (*  hy  the 
If  of  thethirdfihat  a  right  line  drawen  from  two  pointes  taken  in 

Cc.ij.  the 


Second  path 


7  hird parti 


CoftJirH&htt. 


Vmonftra- 

mn. 


An  addition 
ofPtlttmns. 


the  circumference  of  a  circle,  (hall fall  within  the  circle.  Which 
was  required  to  be  demonjlrated, 

5 ^The  2.  Trobleme.  The  17.  Tropoftion0 

From  a poyntgeuenjo  draw  a  right  line  which  Jhall touch  a 
circlegeuen • 

j  Tppofe  that  the  poyntgeuen  he  jf,and  let  the  circlegeuen  he  SOD. 
It  is  required  from  the poynt  .A  to  draw  a  right  line  "Which Jhall  touch 
the  circle  BC  D  .  Take  (by  the  firjl  of  the  third)  the  centre  of  the 
circle }and  let  the  fame  be  F.  And(by  thefrfl petitionjdraw  the  right 
line  AIDE.  And  making  the  centre  F,  and  the  space  ylF,  defcrihe  ( hyy  third 
petition )  a  circle  .AF G.  And from  the  poynt  D  raife  1 ?p( by  the  1 1  *of  the firjl ) 
nto  the  line  E  A  a  perpendicular  line  D  F.  And  (by  the  firjl  petition )  draw# 
thefe  lines  EftF  and  A  ft  .  Then  I 
Jay,  that  from  the  point  A  is  drawen 
to  the  circle  BCD  a  touch  line  A  ft. 

For  for  of  much  as  the  point  E  is  the 
centre  of the  circle  BCD ,  and  alfo  of 
the  circle  A  F  Gyherfore  the  line  F  A 
is  equall  to  the  line  E  F,  and  the  line 
F  D  toy  line  F  ft,  for  they  are  drawen 
from  the  centre  to  the  circumference. 

Wherefore  to  thefe  two  lines  AF  and 
E  ft,  are  equall  thefe  two  lines  E  Fir 
F  D,  and  the  angle  at  the poynt  F  is 
common  to  them  both :  Wherefore  (  by 
the  4 .  of the  firjl)  the  baje  DF  is  equall  to  the  bafe  A  ft,andy  triangle  DEF 


But  the  angle  ED  F  is  a  right  angle :  Wherfore  alfo  the  angle  Eft  A  is  a  right 
angle ,  and  the  line  F  ft  is  drawen  from  the  centre .  But  a  perpendicular  line 
drawen  from  the  end  of the  diameter  of a  circle, toucheth  the  circle  (byy  CoreUa» 
ry  of  the  \6.of the  third  ) .  Wherefore  the  line  A  ft  toucheth  the  circle  BCD. 
Wherfore from  the  point  %euen, namely,  A,  is  drawen  Tfntoy  circle  gene  ft  C  D9 
a  touch  line  yl  ft :  "Which  "Was  required  to  be  done. 

jfy  addition  offtelitarius , 

Vnto  a  right  lyne  which  cutteth  a  circle,  to  drawe  a  parallel  line  which  lhaK 
touch  the  circle, 

*  Suppofe 


es*  Folpfe 

Suppofethat  .the  right  lyije.  AB  do  cut  the  circle  AB  Cinthepoyfites  AandB*  It  is 
require^ todrawe  vntpthelineAB  a  parallel  lyne  &  F  H 

which  ihall  touche  the  circle.'  Let  the  centre  of  the 
circle  be  the  point  D.  And  deuide  the  lyne  A  B  into 
two  equall  partes  in  the  point  E.  And  by  the  point 
E  and  by  the  centre  D,  draw  the  diameter  CDEF, 

And  from  the  point  f'(whlch  is  the  ende  ofthe  dia¬ 
meter)  ray  fe  vpf  by  the  1 1 .of  the  fir  if)  vnto  the  dia¬ 
meter  Cl  F  a  perpendicular  line  G  F  H.  Then  I  fay 
rthatthe  lyne  G  F  H  (which  by  the  correilary  of  c he 

1  <?«.of  this  booke  toucheth  the  circle)  is  a  parallel 
vnto  the  line  A  B.For  forafinuch  as  the  right  line  C 
Ffallyng  vpon  either  of  thefe  lines  AB  &  GHma- 
keth  all  the  angles  at  the  pbint  E'  right  angles  (by 
the  i  ,of  this  bokejand  the  two  angles  at  the  point 
F  are  luppofed  to  be  right  angles :  therforef  by  the 

2  p,of  the  firft)  the  lines  A  B  and  G  H  are  parallels :  which  was  required  to  be  done.And 
this  Probleme  is  very  cominodious  for  the  inscribing  or  circumfcribing  of  figures  ifl 
or  about  circles. 


cl he  1 6,  Theorem^ 


This  Pn- 

btems  cimw-fc. 
Amu  fa?  %m 
in  farthing  c.-A 
ctrtumjen- 
Hngofpgur-n 
m  or  about 
sirties* 


onion* 


.v  .  ,,  •  f 

ffa  right  lyne  touch  a  circle, and  from  the  centre  to  the  touch 
be  dr  amen  aright  line/ that  right  line fo  dr  amen Jhalbe  a per* 
pendicular  lyne  to  the  touchelyne* 

jfippofe  that  the  right  line  ID  F  do  touch  the  circle  ABC  in  the  point  ‘Dmonfln- 
And  take  the  centre  o  f  the  circle  ABC,  and  let  the  fame  be  F.And  mn  . 
fffgfh  the  jiff  petition )  from  the  poynt  F  to  the poynt  C  dr  awe  a  right 
^  dme  FC .  T  hen  I fay, that  C  Fisa  perpendicular  line  to  D  EcFor  if 


J  s 

not  ,draw(hy  the  i  z.of the  ftp) from  the 
poynt  F  to  the  line  D  E  a  perpendicular 
line  EG.  .. And  for  a fmuch  as  the  angle 
FGC  is  a  right  angle,  therefore  the  angle 
GCF  is  an  acute  angle:  Wherefore  the 
angle  FG  C  is  greater  theny  angle  FCG, 
hut  Imto  the  greater  angle  is Jubtended 
the  greater  [ide(  by  the  19.  ofthe  firft). 

Wherefore  the  line  FC  is  greater  then  the 
line  F  G.  'But  the  line  F  C  is  equall  to  the 
line  F'Bjfor  they  are  dr awen  from  the 
centre  to  the  circumference:  Wherfore  the 
line  FB  alfo  is  greater  then  the  lineFG, 
namely /he  kffe  then  the  greater.^hich  is  impojlihle.  Wherefore  the  line  EG  is 
not  a  perpendicular  line  lento  the  line  D  F.And  in  like fort  may  K>e  proue/hat 
no  other  line  is  a  perpendicular  line  Imtoj  line  D  F  bejides  the  line  F  C:  Where¬ 
fore  the  line  FC  is  a.  perpendicular  line  to  DE.  If  therefore  a  right  line  touch 
jn  - '  *  '  ■'  -  Cc.iij. 


An  other  de- 
monflration 
after  Qrsn~ 
tint. 


^Iheihird^Booltp 


4  circle  ^frorfiy  centre  to}  touch  be  drawena  right  line  y  right  line fo  dr  amen 
fall  be  a  perpendicular  line  toy  touch  line :  lohich  Teas  required  to  be proued. 


f  An  other  demonflration  after  Orontius . 

Suppofe  that  the  circle  geuen  be  A  B  C,  which  let  the  right  lyne  D  ton  ch  in  the 
point  C»  And  let  the  centre  of  the  circle  be  the 
point  F.And  draw  a  right  line  from  F  to  C.Then 
I  fay  that  the  line  FC  is  perpendicular  vnto  the 
line  DE. For  if  the  lineF  C  be  notaperpediculer 
vnto  the  line  D  E,  then, by  the  conuerfe  ofthe  x. 
defi  nition  of  the  firft  boke,the  angles  D  C  F  &  F 
C  E  fbal  be  vnequalb&:  therfore  the  one  is  grea¬ 
ter  then  a  right  ang!e,and  the  other  is  lelfe  then 
a  right  angle ,  (Tor  the  angles  DCF  and  F  C  E 
are  by  the  x  3  .of  the  firft  equall  to  two  right  an¬ 
gles)  Let  the  angle  FCE,ifit  be  poffible,be  grea¬ 
ter  then,  a  right  angle,  that  is,  let  it  be  an  obtufe 
angle.Wherfore  the  angle  DCF  fhal  be  an  acute 
angle.  And  forafmuch  as  by  fuppofitio  the  right 
lineD  E  toucheth  the  circle  AB  C ,  therefore  it 
cutteth  not  the  circle.  Wherefore  the  circumfe¬ 
rence  B  C  falleth  betwene  the  right  lines  DC&CF:&  therfore  the  acute  and  re&iline 
angle  D  C  F  (hall  be  greater  then  the  angle  ofthe  femicircleBC  F  which  is  contayned 
vnder  the  circumferece  B  C  8c  the  right  line  C  F.And  fo  fhall  there  be  geue  a  re&iline  & 
acute  angle  greater  then  the  angle  of  a  femicircle:  which  is  contrary  to  the  1 6 ,  propo¬ 
rtion  of  this  booke.Wherfore  the  angle  D  C  F  is  not  leffe  then  aright  angle, In  like  fort 
alfo  may  we  proue  that  it  is  not  greater  then  a  right  angle.Wherfore  it  is  a  right  angle, 
and  therfore  alfo  the  angle  F  C  E  is  a  right  angle.Wherefore  the  right  line  F  C  is  a  per¬ 
pendicular  vnto  the  right  line  D  E  by  the  10. definition  of  the  firft ;  which  was  required 
to  be  proued.  \  _•  >  - 

^The  \  jfTheoreme .  fhe\<y.  Tropoftion. 

ffa  right  lyne  doo  touche  a  circle,  and  from  the  point  ofthe 
touch  he  rayfed  vp  vnto  the  touch  lyne  a  perpendicular  lyne, 
rath  at  lyne Jo  rayfed vp  is  the  centre  ofthe  circle . 


Demonftra - 
tion  leading 
to  an  impo/0 
fbilkie. 


rife  that  the  right  line  D 
| touch  the  circle  ABC  in 
j  C,  And  from  C  raife'Tpfyy  n.oj 
the  frjlfnto  the  line  IDE  a  perpendicu* 
lar  line  C  A.  Then  I  fay  jth at  in  the  line  3 
C  A  is  the  centre  of the  circle .  For  if  not, 
then  fit  be  pofible }  lety  centre  be  Trith* 
out  the  line  C  A  }as  inj poynt  F  And  ( by 
the  firjl petition)  draw  a  right  line  from  C 
to  F.  And  for  afmuch  as  a  certaine  right 
line  D  E  toucheth  the  circle  A  ©  C?and 
from  the  centre  to  the  touch  is  drawena 
right  line  C  l\  therefore  ( by  the  i8»  ofthe  third)  F  C  is  a  perpendicular  line  to 

\  •  '  •  '  '  DE . 


ofSuclides  Elementes *  Fol,$6* 

0  E.  Wherefore  the  angle  FCEis  a  right  angle .  Eut  the  angle  jfCE  isalfo 
aright  angle:  Wherefore  the  angle  EC  E  is  equal!  to  the  angle  A  CE, namely s 
the  lejfe  Imto  the  greater :  yohich  isimpofible  Wherefore  the  poynt  F  is  not  the 
centre  of the  circle  A  E  C,  And  in  like  fort  may  ypeproue  ,y  it  is  no  other  Cohere 
hut  in  the  line  A  C .  If  therefore  a  right  line  do  touch  a  circle, and  from  the  point 
of the  touch  he  raifed  Ipp  Imto  the  touch  line  a  perpendicular  line ,  in  that  line fc 
rat  fed  lop  is  the  centre  of the  circle :  y&hich  % vas  required  to  he  proued , 

f^pThe  18,  clheoreme%  TheiQ.fPropoJitwn. 

In  a  circle  an  angle  fet at  the  centre Js  double  to  an  angle  fet 
at  the  circumference  ,fo  that  both  the  angles  haue  to  their 
bafeoneand  the  fame  circumference. 

Vpfofe  that  there  be  a  circle  ABC,  and  at  the  centre  thereof , namely } 
the poynt  EJety  angle  EEC  he  fet,&  at  the  circumference  let  there 
be  jet  the  angle  E  A  C,  and  let  them  both  haue  one  and  the fame  bafe, 
namely, the  circumference  EC,  T  hen  I fay,  that  the  angle  EEC  is 
double  to  the  angle  E  AC,  0rawj  right 
line  A  E,and(by  the J'econd  petition  fx* 
tend  it  to  the  poynt  F.  Flow  for  afmuch 
as  the  line  A  E  is  e quail  to  the  line  E  E, 
for  they  are  drawen from  the  centre  lento 
the  circumference ,  the  angle  EAE  is  e*  ^ 
quail  to  the  angle  EE  A(  by  the  S  *  of the 
firfl  ) .  Wherefore  the  angles  EAE  and 
E  E  yf  are  double  to  the  angle  EyfE, 

Eut  (  by  the  Z1*  of  the  fame  )  the  angle 
EEF  is  equall  to  the  angles  E  jdE  and 
E E yf:  Wherefore  the  angle  EEF  is 
double  to  the  angle  E  A  E .  And  by  the 
J'ame  reajon  the  angle  F  EC  is  double  to  the  angle  E  ylC.  Wherefore  the  “tohole 
angle  EEC  is  double  to  the  ivbole  angle  E  A  C, 

KAgaine  fuppofe  that  there  he  fet  an  other  angle  at  the  circumference  ,  and  let 
the Jame  be  EE)  C.  And(by  the firft  petition)draw  a  line  from  0  to  E.  And  (by 
the fecond  petition)extend  the  line  0Elonto  the  poynt  G .  yfnd  in  like fort  may 
ice  proue,that  the  angle  G  EC  is  double  to  the  angle  E  0  C.  Of  yohich  the  an * 
gle  G  EE  is  double  to  the  angle  E0E.  Wherfore  the  angle remayning  E  EC 
ts  double  to  the  angle  remayning  E0  C.  Wherfore  in  a  circle  an  angle  Jet  at  the 
centre, is  double  to  an  angle  fet  at  the  circumference  fa  that  both  the  angles  haue 
to  their  hafe  one  and  the  fame  circumference :  lehich  yeas  required  to  be  demon* 
sirated, 

'  Ccfnij.  z’&  'Tbe 


T Wo  cafes 
in  thys  Pro* 
f  option  the 
one  when  the 
angle  jet  at 
the  circumfe - 
renceinclu- 
deth  the  cen~ 
ter* 

VemonfttA- 
mn . 


The  ether 
wbe  the  fame 
angle  fet  at 
the  circumfe - 
renccinclu- 
deth  not  the 
center* 


Irhff  hc  ip.  'Theorems.  fths  zi.  TroPofitwn. 

'  vj~  "  -  ‘  •'■v  •  •''  '  ■  ■  -  :  l  .<  V-'V-i  v%-'*yv.V'  i! 

In  a  circle  the  angles  Tvhich  conjlsl  in  one  and  the  felfe  fame 
JeBlon  or  figment ,are  equall  the  one  to  the  other , 

‘ \  ’  '  v- *  '  4  w  •/-.  Mr  1  V  “  *  *  •;*  a,**“->* 

:  :  o  i  \  .  •••■’■  .C.<;C  -  A  ?  ■: ;  ; 

Vppojey  there  be  d  circle  A3  C  Dps  in  the  fegment  therof'B  A  E  D, 
let  there con ftfi  thefe  tingles  3  AID  and  3  ED .  Then  I fay,  that  the 
angles  3  AD  and  3  ED  are  equall  the  oneto  the  other. Take(by  the 
fir  ft. of  the  third  )  the  centre  of  the  circle 
Conjlmtm*  yd  3  CD,and  let  the  fame  be  the  point  F. 

Andfhy  thefirfipeiition)drct\v  theje  lines 
pemon  first*  <J$p  anp  FT),  ISLow  for  afinuoh  as  the 
angle  3FD  is fet  at  the  centre,  and  the 
angle  3  AD  at  the  circumference ,  and 
they  bane  both  one  andy fame  bafeyiame* 
lyphe  circumference  3C  D  .therefore  the 
angle  3  FD  is  (by  the  Tropofition going 
before  )  double  to  the  angle  3  AD :  and 
by  the fame  reafon  the  angle  3FD  is  aU 
fo  double  to  the  angle  3  ED.  Wherefore 
(  by  the  7*  common  fentence  )  the  angle 
3  jdD  is  equall  to  the  angle  3  ED  .Wherefore  ina  circle  the  angles  “Which 
cohfifte  in  one  and  the  felfe  fame  fegment yire  equall  the  one  to  the  other  :  1 vhich 
"Was  required  to  be proued. 


Three  cares  m 
this  Propefi- 
tion. 

The firti  cefe. 
The  fecond 

uje. 


1b  this  proportion  are  three  cafes.For  the  angles  confifting  in  one  and  the  felf  fame 
fegnienr,the  fegment  may  either  be  greater  the  a  femicircle,  or  lefle  then  a  femicircle, 
orelsiullafemiciycle^Forthe  firfl  cafe  the  demonftration  before  put  ferueth. 


But  now  fuppofe  that  the  angles  BzAD 
and  B  E  Dd  o  confift  in  the  fe&io  BAT),  which 
let  be  leffe  then  a  femicircle. Euen  in  this  cafe  al¬ 
io  I  fay  that  the  angles  2?  A  D  and  B  ED  are  e- 
quail. For  draw  a  right  line  from  A  to  E.  And  let 
the  lines  A  D  and  BE  cutte  the  one  the  other  in 
thepoynt(7,  wherefore  the  fegment  cACEis 
greater  then  a  femicircle.  And  therfore  by  the 
firfl  part  of  this  propofition  the  angles  whiche 
are  in  it,  namely,  the  angles  %A  T  E  and  EDA 
are  equall  the  one  to  the  other.  And  forafmuch 
as  in  the  triangle  eA  B  G  the  inward  and  oppo¬ 
site  angles  (7  and  Cj  A B  are  equall  to  the 
outwarde  angle  B  G  D,  and  by  the  fame  reafon 
the  two  angles  E  D  (7  and  G  E  D  of  the  triangle 
D  EG  are  equall  to  the  felfe  fame  outward  angle 
B  GD.  Wherfore  the  two  angles  A  B  G  and  G  A 
B  are  equall  to  thc-Wo  angles  ED  G  and  G  ED 


by  the 


Fo/.p  7. 


by  the  firft  commofentenec.From  which  if  there 
be  taken  equall  angles,  namely,  j4B  G ,  and E <D 
G}  the  angle  remainyngZf-^G’  (hall  be  equall  to 
the  angle  remayning  D  E  G, that  is,theangle  B  A 
D  to  the  angle  D  E  B( by  the  third  common  fen- 
tence)  which  was  required  to  be  proued. 

The  felre  fameconftru&ion  and  dcmonftrati-  ^ 
cn  will  alfo  ferue,if  the  angles  were  fet  in  a  femi- 
circle  as  it  is  playue  to  feejn  the  figure  here  fet. 


0 


The  third s&fe* 


fr&The  lo.Theorcme,  The  21*  Trope  fit  ion. 

If  within  a  circle  be  deferibed  a  figure  of  former  fides,  the  an¬ 
gles  therof  which  are  oppofite  the  one  to  the  other >  are  equall 
to  two  right  angles . 


f  ^PP°fe  ^?at  there.  a  circle.  ABC  ID, and  let  there  he  deferibed  in  it 
i^nX^  a  figure  of  fiver fides , namely ,  A  B  CD.  T  hen  I fay ,tbat  the  angles 
thereof  finch  are  oppofite  the  one  to  the  other, are  equall  to  two  right 
■ — | —  'angles.  Draw  (by  the firft  petition )  thefe  right  lines  A  C  and  B  D.  Conjhufih® 
Tdow  for  ajmuch  as  (by  the  3  i*  of the  firft)  the  three  angles  of euery  triangle  are 
equal l  to  two  right  angles:  therforey  three  angles  of  the  triangle  ABC,  namely .  Demon 
CAB,ABCyandBCA,areequaUto  ***. 

two  right  angles .  But  (by  the  zi.  of  the 
third )  the  angle  CAB  is  equall  to  the 
angle  B  DC  for  they  conffi  in  one  and 
the felf fame fegmet,  namely,  B  ADC. 

And  (by  the  fame  Tropofition)  the  an* 
gleACB  is  equall  to  the  angle  A  DB, 
for  they  conffi  in  one  arid  the  fame  fig * 
merit  ADC  B  .  Wherefore  the  K>ho/e 
angle  AD  C  is  equall  toy  angles  B  AC 
and  ACB: put  the  angle  ABC  com * 
monto  them  both.  Wherefore  the  angles 
A  B  C,B  A  C,  and  A  C  B,are  equall  to 
the  angles  ABC  and  ADC.  But  the 

angles  ABC,  B  AC,  and  A  C  B,  are  equall  to  two  right  angles .  Wherefore 
the  angles  ABC  and  ADC  are  equall  to  two  right  angles .  And  in  like fort 
alfo  may  ive  prone, that  the  angles  BAD  and  DCBare  equall  totworight 

angles: 


’Demonfird- 
tion  leading 
to  an  impof- 
fibilme . 


An  addition 

cf  Cam  pane 
dsmonjlrated 
by  Pdsiarms. 


"Demonjlra 
tivn  leading 
to  an  impofii- 
bilitts , 


-  '  ^  i  f  J(v  - 

.  2/  therefore  ‘toitbin  a  circle  be  defcrihed  a  figure  offower fides,  the  an* 
gles  thereof  ~%>btcb  are  oppofite  the  one  to y  other,  are  equal!  to  two  right  angles: 
Tvhich  t>as  required  to  he proued. 


>-&'! he  zuT heoreme ,  The  1 3 . Tropofition . 

cOpon  one  and  the  fife  fame  right  line  can  not  be  defcrihed 
Wo  like  and  vne  quail fegmentes  of circles  falling  both  on  one 
and  the felfe fame  fide  of  the  line. 


Or  if  it  he  pofiihlejet  there  he  defcrihed  lapon  the  right  line  A  B 
two  like  zjr  Imequall fe Elions  of  circles , namely ,  ACB  iy  ABB, 
falling  hoth  on  one  and  the  felfe  fame  fide  of  the  line  J  B .  And 
(by  the  fir  ft  petition )  drawe  the  right  line  JC  B,  and(  by  the 
third  petition)  drawe  right  lines  from  C  to  B,  and  from  B  to  B. 
.And  for  afinuch as  the figment  ACB 
is  like  to  the  figment  jC B  B :  and  like 
fegmetes  of  circles  are  they  dnch  haue 
equal l  angles  (  by  the  io,  definition  of 
the  third).  Wherefore  the  angle  A  CB 
is  e quail  to  the  angle  ABB ,  namely , 
the  outward  angle  off  triangle  CBB 
to  the  in  ward  angle :  ~$>bicb(  by  the  1 6* 
of thefirft)  is  imqmfiihle.Wherfore  lop* 
on  one  and  the  felf  fame  right  line  can  not  he  defcrihed  two  like  ty  Unequal! peg* 
mentes  of  circles  falling  both  on  one  ty.  the felfe  fame fide  of the  line:  T vhich  "tyas 
required  to  be  demonjlrated. 

Here  Campane  addeth  that  vpon  one  and  the  felfe  fame  right  lyne  cannot  be 
deferibed  two  like  and  vnequall  lections  neither  on  one  and  the  felfe  fame  fide  of 
the  lyne,nor  on  the  oppofite  fide.That  they  can  not  be  delcribed  on  one  and  the 
felfe  fame  fide3hath  bene  before  demonftrated3and  that  neither  alfo  on  the  oppo¬ 
fite  fide3Pelitarius  thus  demonftrateth. 


V 


A  B 


.  Let  the  fe&ion  A  B  C  be  fet  vppon  the  lyne  'A  C,and  vpon  the  other  fide  let  be  fee 
the  fe&ion  ADC  vppon  the  felfe  fame  lyne  A  C, 
and  let  the  fe&ion  ADC  belyke  vnto  the  fe&i¬ 
on  ABC.  Then  I  fay  that  the  fe&ions  A  B  C  and  A 
D  C  being  thus  fet  are  not  vnequal.Forifit  be  pof- 
fible  let  the  fe&ion  A  D  C  be  the  greater.  And  de- 
uide  the  line  A  C  into  two  equal  partes  in  the  point  A 
E.  And  draw  the  right  lyne  BED  deuiding  the  lyne 
A  C  right  angled  wife.  And  draw  thefe  right  lynes  A 
B,  C  B3A  D  and  C  D,  And  forafmuch  as  the  lection 
A  D  C  is  greater  then  the  fe&ion  A  B  C,  the  perpen 
dicu.lar  lyne  alfo  E  D  (hall  be  greater  then  the  per¬ 
pendicular  lyne  E  B  :  as  is  before  declared  in  the 
ende  of  the  definitions  of  this  third  booke*  Wher- 


ofSuclides  Elemmtem  Foh  p8 * 

fore  from  the  ly  ne  E  Dgut  of  a4y  netiquaj.l  tp  the  Iyne  E  B  s  which  let  be  E  F.  And  draw 
tfiefe right  iynes  A  F  and  C  F.Now'  then(by  the  4*of the  firlljthe  triangle  A  E  B  (hall  be 
ecpiall  to  the  triangie.A  E'F,and  the  angle  E  B  A  iliall  be  equall  to  the  angle  E  FA.  And 
by  the  fame  reafon  the  angle  E  B  C  (hall  be  equall  to  the  angle  E  F  G  .  Wherefore  the 
whole  angle  A  B  C.  is-equail  to  the  whole  angle  A  F  C.But  by  the  z  i.  ofthefirfigthe  an¬ 
gle  A  F  C  is  greater  then  the  angle  A  D  C.Wherfore  alfo  the  angle  A  B  C  is  greater  then 
the  angle  A  DC.  Wherefore  by  the  definition  the  fedtions  AB  Cand  AD  Care  not 
iyke,which  is  contrary  to  the  fuppofition.  Wherefore  they  are  not  lyke  and  vnequall ; 
which  was  required  to  be  proued. 


S^The  iiLTheorme ,  TThe  i^SPropoftion. 

Like fegmentes  of  circles  defcriledvfipon  equall  right  lines, 
are  equall  the  one  to  the  other . 


Wppofe  thatypon  thefe  equall  right  lines  AB  and  CD  he  deferibed 
P thefe  like  fegmentes  of  circles /Lamely >  AF  B  and  CFO).  Then  I fqy} 
^  that  the figment.  ABE  is  equall  to  the  ferment  CFD  .  For  putting 
the fegment  A  E  B  'Opon  the  figment  C  FDynd  the  poynt  A'Opon  y  pojnt  C} 
and  the  right  line  A  B  hpon  the  right  line  C  D }  the  poynt  B  alfo  (hall fall  Tvpon 
the  poynt  Dfiory  line  A  B  is  equall  to  the  line  C  D.And  the  right  line  AB  ex> 
aFily  agreing  Soith 

the  right  line  CDfi  E  F 

fegment  alfo  AEB 
Jhall  exaFlly  agree 
"S nth  the  fegment 
CFD  .  For  if  the 
right  line  AB  do 
exactly  agree  loith  /i 
the  right  line  C  D , 

andthe figment  AEB  do  not  exaFlly  agree  *t»ith  the figment  CFD?  but  dif¬ 
fer  eth  as  the figment  CGD  doth:  Now(by  the  i  of  the  third)  a  circle  cutteth 
not  a  circle  in  more  pointes  then  two  fit  the  circle  CGD  cutteth y  circle  C  FD 


in  more  pointes  the  two  y  hat  is  yn  the  points  Cfiynd  D:K>hich  is(by  the  fame ) 
impofiible  .Wherefore  the  right  line  AB  exaFlly  agreing  leith  the  right  line 
C  Dy  the  fegment  AEB  Jhall  not  hut  exactly  agree  hath  the  fegment  CFD: 
Wherefore  it  exaFlly  agreeth  K>ith  and  is  equall  'Onto  it .  Wherefore  like  fig* 
mentes  of  circles  defer ibed'Opon  equall  right  lines ,are  equall  the  one  to  the  other: 

‘ Hebich  "Mas  required  to  be proued. 


ThisFropofition  may  alfo  bedemonftrated  by  the  former  propofition.  For  if  the  fe- 
&ions  AEB  and  CFD  being  like  and  fet  vpon  equall  right  lines  esf  B  and  C‘Z>,  ihould 
be  vnequailythen  the  one  beyng  put  vpon  the  other, the  great;  r  iliall  exceede  the  leife: 
but  the  line  A  A  is  one  line  with  the  line  C'Di  Cq  that  tlierby  ihalfolloiy  the  contrary  of 
the  former  Propofition. 

an  A  '  Belitarim 


Demonftra-* 
tion  leading 
loan  impoF 
fibiline. 


An  other  de-* 
monjiration. 


*An  other  /#- 

monjlrAtio 
$er  Fslitmnu 


Conjlmtlkn, 


three  cafes  in 
skis  Frapo/i- 
tiott. 

the prft  cafe, 

V  V ,  '  v, 

DymmUrA- 

Sion* 


Belitarius  demonUrateth  this  Proportion  an  other  7$  ay. 

Suppofe  that  there  be  two  right  lines  tsf B  ScC  D  which  let  be  equall  .•  and  vpon  the 
let  there  be  fet  thefelike  fcftions  A  B  /^and  C  D  E.yhcn  I  fay  that  the  faid  fedions  are 
equall.For  if  no  when  let  C  E  D  be  the  greater  fedion.  And  deuide  the  two  lines  *A  B 
and  CDinto  two  equall  partes,thcline  A 
Bm  the  pointed,  and  the  line  CD  in  the 
point-G’.And  cred  two  perpendicular  lines 
F  ifand  G  A.Anddraw  thefe  right  lines  A 
XdcKB;  EC}ED.  Andforafrauchasthe 
fedion  CAD  is  the  greater,  therefore  the 
pcrpendicularline  GDIs  greater  then  the 
perpendicular  F  if:  From  the  lyncG  E  cut 
of  a  line  equall  to  the  line  fifjwhich  let  be 
G  H :  and  draw  thefe  right  lines  C  i/and  H 
DfAnd  forafmuch  as  in  the  triangle  *A  IC 
F  the  two  iides  A  F  and  F  K  are  equall  to  the  two  fides  CGScG  H  of  the  triangle  CH~ 
G,and  the  angles  at  the  pointes  A  and  G  are  eq  ual  ( for  that  they  are  right  angles)  thcr- 
forefby  the  4. of  the  firft)the  bafe  K  is  equall  to  the  bale  CH, and  the  angle  A  K F  to 
the  angle  CHG.And  by  the  famereafon  the  angle  B  K  Fis  equall  to  the  angle  D  H  G. 
Wherfore  the  whole  angle  A  KB  is  equall  to  the  whole  angle  C H  D.But  the  angle  CH 
Dis  greater  then  the  angle  C  ED  by  the  2 1  .of  the  firft*Wherfore  alfo  the  angle  A  KD 
is  greater  then  the  angle  C E  D.  Wherfore  the  fedions  are  not  lyke,  which  is  contrary 
to  the  fuppofition. 

•4 

The  3 .  Trobleme .  The  25.  Tropofition . 

ctJ figment  of  a  circle  beynggeuen  to  deferibe  the  whole  cir > 
c/e  of  the  fame  fegment . 

Vppofe  that  y figment geuen  be 
ABC.lt  is  required  to  dejerihe 
the  yohole  circle  of  the fame fig* 
ment\AB  C.  Deuide  (by  the  10.  of  the 
frjljtbe  line  A  C  into  two.equall  partes 
m  the poynt  D .  And  (by  the  1  u  of  the 
fame) from  the  poynt  D  raife  yp  ynto 
the  line  A  C  a  perpendicular  line  B  D. 

And  ( by  the frit  petition)  draw  a  right 
fine from  A  to  \BJSLow  then  the  angle  ABD  being  compared  to y  angle  rB  A  (Dp 
is  either  greater  then  itfir  equall  lento  ityr  lejfe  then  it. 

First  let  it  he  greater,  And  ( by  the  23  ♦  of  the ftmeftpon  the  right  line  B  As 
and  ynto  the  poynt  in  it  A,  make  ynto  the  angle  ABD  an  equall  angle  B  A  E. 
And(by  the fecond  petition)extend  the  line  B  D  ynto  the  poynt  E.  Andfby  the 
fir ft petition)  draw  a  line from  E  to  C.  Nora  for  afinuch  as  the  angle  ABE  is  e* 
quail  to  the  angle  BAE,  therefore  (by  the  6  ♦  of  the firfi)the  right  line  EB  is 
equall  to  the  right  line  A  E .  And  for  afinuch  as  the  line  AD  is  equall  toy  line 
D  C  and  the  line  D  E  is  common  to  them  both :  therefore  thefe  two  lines  A  2) 

and 


of  Suclides  Elemerites * 


FoLpp, 


and DE}  are  equall  to  thefe  two  lines  C ID  and  D  E  the  one  to  the  other .  And 
the  angle  .A  D  E  is  ( by  the  4  *  petition')  equall  to  the  angle  C  D  Ey for  either  of 
them  is  a  right  angle .  JV here  fore  ( by  the  4 .  of the firft  )  thebafe  AE  is  equall 
to  the  bafe  C  E .  'But  it  is  proved  ghat  the  line  A  E  is  equall  toy  line  B  E.Wber* 
fore  the  line  BE  a  jo  is  equall  to  'the  lineC  E.  Wherefore  thefethree  lines  A  Ey 
E  Byand  E  C  yare  equall  the  one  to  the  other  .  Wherefore  making  the  centre  E? 
and  the  space  either  A  Ey  or  EByor  E  C:  defcribe  (by  the  third  petition)  a  cir* 
demand  it  fallpqffe  by  the  poyntes  Ay  By  C .  Wherefore  there  is  defcribed  the 
ivhole  circle  of the  fegment  gemn..  And  it  is  mamfeffthat  the fegment  ABC 
is  lejfe  then  a  femicircle for  the  centre  Efalleth  without  it* 

Tlx  like  demon f  , ration  dlfo  Will 
feme  if  the  angle  ABD  be  equall  to 
the  angle  B  AD  .  For  the  line  A  D 
being  equall  to  either  of  the fe  lines y 
B  Dy  and  D  Cy  there  are  t  hree  lines y 
D  A yD  By.ndD  Cy  equall  the  one  to 
the  other .  So  that  the  point  D fall  be 


B 


ry/ri  • 

i’  ^ 

...  XV, 

■  ;  “4 

j  / 

’  \ 

Thefctoni 

Safe* 


and  ABC  fall  be,  a  femicircle.  , 

But  if  the  angle  ABD  be  leffe  then  the  angle 
B  AD  y  then  ( by  the  2t,*of  the  firfi)  t>pon  the  right 
Me  B '  Afnidisnio  the point  in  it  A }  make  TpMo  the 
angle  ABD  an  equall  angle  withiny fegment  A  B  C. 
Andfo  the  centre  of  the  circle  f  allfall  my  lineD  By 
and  it  fall  be the point  E:  and  the  fegment  ABC 
greater  then  a femicircle  Wherefore  a fegment 


the  third 
cafe. 


beinggeuen yhere  is  defcribed  the  whole  circle  of  the  fame  fegment ;  which  Was 
required  to'be  donfty^ 


noj 

of 


X"; 


,  $&{ Corollary*  1 

,  »  flic  '■  nsriT  .  I 

Hereby  angle 

is  equall  to  the  angle  2)  BfJ4ihttt  in  ajfeQionleJie  thenajetnicir - 
dSrkh  afimkfcelejtuffta&r. 

.  .nob  %Q i 

;Thereis  alfoaJTother  generall-w'-aytofindcputthe' 

fore  faid  centre,  which  will  feme  Indifferently  for  any 
faction  vvhatfoeuer.-  And  thafis  tfius.  '^Taicein  the  cir¬ 
cumference  geuon  or  {eaion  ^AC’fhree  pointes  at  all  v  ,  .  g 

sueiitures  vvhichller  be  A,B,C\  And  draw  thefe  lines  A 
B  and 'A  C(by  the  firft  peticion)'  Andf  by  the  1  o.ofthe 
firft)  deuide  into  two  equall  partes  either  of  the  fayde 

iines^he  line  a^Sinthe  point  ©p&sthfelii®ei^Cjn  tie 

point  E.  And  (by  the  11.  ofthefirfl^ifrd'Orth^pointgB^.-'u ' 

Dd.joj  D  and 


id'N.-  v  f 


Another 
more  ready 


Utmonfirfr 


An  ad  dition* 


An  other  eon- 
jlructionand 
demotiflretion 
of  this  Propo¬ 
rtion  after , 
Campari  e 


t&w 


©  and  £  rayfe  vp  vnto  the lines  and  BC perpendicular  lyrics  *& F  and  iZF.  Map 

forafmuch  as  either  of  thefe  angles  B  D  £,and£  EF  is  a  right  angle, a  right  line  produ¬ 
ced  frotn  the  point  D  to  the  point  £,  lhall  deuide  either  of  the  faid  angles :  and  foraf- 
much -as  it- fallefh  vppon  the  right  lines  't>  F  and  %  F,it  fhall  make  the  inward  angles  on 
one  and  the  felfe  fame  fide,  namely,  the  angles  2)  EF 
and-£  i?  F.leife  then  two  right  angles.  Wherefore  fby 
the  fife  peticipn)  the  lines  D  F  and  £  F  being  produced 
lhallconctirre*  Let  them  concurre  in  the  point  F.  And 
forafmuch  as  a ■  certain e  right  line  D  F  deuideth  a  cer- 
taine  right lyneexfF  into  two  equal!  partes  and  per¬ 
pendicularly,  therfore  ( by  the  corollary  of  the  firftof 
this  booke)  in  the.  line  D  F  is  the  centre  of  the  circle,&: 
by  the  fame  reafon  the  cen  tre  of  the  felfe  fame  circle  A 
lhalbe  in  the  right  line  £  F.Wherfore  the  centre  ofthe  . ,  „ 

circle  wherof  ABC  is  a  feftion,is  in  the  point  F,which  is  comrmo  fo  either  of  the  lines 
'D  Fand  £  F.  Wherfore  a  feftion  of  a  circle  being.  geue,namely,the  fedtion  ATS  Cthere 
isdefcnbedthecircleofthe  fame  fedion:  which  was  required  to  be  done. 

And-bv  this  laft  generall  way, if  there  be  geuen  three  pointes,  fet  howfoeuer,  fo  that 
they  be  not  all  three  in  one  right  line,a  man  may  deferibe  a  circle  which  fhall  paffe  by 
all  the  faid  three  pointes.  For  as  in  the  example  before  put, if  you  fuppofe  onely  the  3. 
pointes  A,ByC,  to  be  geuen  and  not  the  circumference  A  B  Cto  be  drawen,yet  follow¬ 
ing  the  felfe  fame  order  you  did  before,  that  is,  draw  a  right  line  from  *A  to  £  and  an 
other  from  B  to  C  and  deuide  the  faid  right  lines into  two  equall  parts,in  the  points  D 
and  £,and  cre<5t  the  perpendicular  lines  D  £and£  Fcutting  the  one  the  other  in  the 
pointF,anddrawaiightline  from  F to £:and making  the  centre  thepointF,  and  the 
Space  F  B  deferibe  a  cirefoan^  it  lhall  pafie  by  . the  pointes  A8cC:  which  may  be  pro-? 
ued  by  drawing  right  lines  fibm-v?  to  F,and  from  F  to  C.  For  forafmuch  as  the  two 
fetes  A  V  zndJ&'F  of  the  triangle  ADF  are  equall  to  the  two  fides  S  TT  and  £>  F  of  the; 
triangle  cB/D\F(Sax  by  fuppofitidn  the  line  tAT) is  equall  to  the  IineJ9£,  and  thelyne, 
£>Fis  cpinmontothem  both)  and  the  angle  A'DFis  equall  to  the  angle  B  D  F  (for 
they  are  both  right  angles  )therfore(by  the  q..of  thefirft Jthe  bafe  AF  ts  equall  to  the' 
bafe  B  F.  And  by  the  fame  reafon  the  line  F  C  is  equal!  to  the  line  F  B.  Wherefore  thefe. 
thred  lines  F  A,F  B  and  F  C  are  equall  the  one  to  the  other .  Wherefore  makyng  the 
centre  the  point.  F  and  the  fpace  F£,it  (hall  alfo  paffe  by  the  pointes  A  and  C ,  Which 
was  req aired  to  be done.This  proposition  fs  very  neceffary  for  many  things  as  you  fhal 
afterward. fee.-  -.;r  .  •  .■  ;  :  ^  '.v^y.  . 

Campane  putteth  all  other  way,  how  to  deferibe  the 
whole  circle  of  a  feftio  geuen .  Suppofe  that  the  fc&ion 
be  ^F.It  is  required  to  deferibe  the  whole  circle  ofthe 
fame  fe&ion.Draw  in  the  fettion  twolmes  at  all  aduen- 
tures  AC  and  B  FLwhich  deuide  into  two  equal! parts 
AC  in  the  point  £, and  B  (D  m  thepointe  F.Then  from 
the  twopotnteS  of  the  deaifion&draw  within  the  fe&i- ' .  ] 

on  two  perpendicular  lines  Ft?. and  F  H  which  let  cutte- 
the  one  the'  , other  in  the.pjmttfc^Mdth-e  centre  of  the 
circle  lhall  bein  either  of  the  faid  perpendicular  lines 
by  the  cdrollairy  of  the  firft  of  this  booke.  Whrirfore  the  p 
cle:  which  was  required  to  be  done. 

But  if  the  lines  EG  8c  FH  do  not  cucthe  one  theo- 
ther, but  make  one  right  line  as  doth  6? Ffinthefecod 
figure:  which  happeneth  when  the  two  lines  AC  and 
B  D  arc  equidiftant. Then  the  line  G H,  being applycd 
to  cither  part  of  the  circumference  geuen,  (hall  paffe 
by  the  centre  of  the  circlc,by  the  felfe  feme  Corollary.. H 
For  thelines  £  G  and  F  H  cannot  be  equidiftant.  For 
then  one  and  the  felf  feme  circumference  ftrould  hauc 
two  centres.Whcrfore  the  line  ALC?  being  deuidedia- 

hi:  to  .bC 


centre  ofthccifw 


l  K 

i;  -  1 

F 

C  T/V' 

'  FoLwoe 

to  two  equal!  partes  in  the  point  Kxhx  faid  point  K  (hall  be  the  centre  of  the  fe&ion. 

Pelkaiius  here  addeth  a  bride  way  how  to  finde  out  the  centre  of  a  circle3which 
is  commonly  vied  of  Artificers, 

Sup  pole  that  the  circumference  be  A  B  C  D,frhofe  centre  it  is  required  to  finde  our. 
Take  a  point  in  the  circumference  geuen  which  let  be  A,  vppon  which  deferibe  a  circle 
with  what  openyngofthecompaileybu  will, which  let  be  EFG.  Then  take  an  other 
point  in  the  circumference  geuen  which  let  be  B,  vpon  which  deferibe  an  other  circle 
.  with  the  fame  opening  of  the  compaffe  that  the  cir- 
cle  EFG  was  ddcribed,-  and  let  the  fame  be  E  H  G , 
which  lec'cut  the  circle  E  F  G  in  the  two  pointes  E 
andG-.f  I  haue  not  hc_re  drawen'the  whole  circles, 
blit  onely  tliofe  partes  of  them  which  cutthe  one 
theother  for  auoyding  of  confufion )  And  drawe 
from  thofe centres  thefe  right  lines  A  E,  B  E,  AG, 
and  B  G, which  foure  lines  lhail  be  equal!,  by  reafon 
they  are  femidiameters  of  equall  circles.  And  draw 
a  right  line  from  A  to  B,  and  fo  lhail  there  be  made 
two  Ifofedes  triangles  A  E  B,and  A  G  B  vnto  whom 
the  line  A  B  is  a  common  bale.  Now  then  denide  the 
line  A  B  into  two  equal  partes  in  the  point  K  which 
rouft  nedes  fail  betwene  the  two  circumference  E  P 
G and  EHG,  otherwife  the partfhould be  greater 
then  his  whole,  Drawe  a  line  from  E  to  K  and  pro¬ 
duce  it  to  the  point  G.  Now  you  fee  that  there  are 
two  Bofceles  triangles  deuided  into  foure  equall 
triangles  E  A  K,  EBK,GAK  and  GBE.  For  the 

two  (ides  A  Band  A  K  of  the  triangle  AEK  are  equall  to  the  two  fides  BE  andBK  of 
thetriangleB  EK5and  the  bafe  EK  is  common  to  them  both.  Wherefore  the  two  an¬ 
gles  at  the  point  Kof  the  two  triangles  A  EKand  B  EKaiebythe  8,ofthe  fir  ft  equall: 
and  therfore  are  right  angles.  And  by  the  fame  reafon  the  other  angles  at  the  poynte  K 
are  right  angles, Wberfbre  E  G  is  one  right  lyne  by  the  14.  of  the  .firft .  Which  foraf- 
much  as  it  deuideth  the  line  A  B  perpendicularly,  therefore  it  pafteth  by  the  center  by 
by  the  corollary  of  the  firft  of  this  booke.  And  fo  if  you  take  two  other  poynte  s,name« 
C  and  D  m  the  cfrcumference  gcuen,and  vpon  the 
deferibetwo  circles  cuttyng  theone  the  other  in 
the  pointes  L  and  M,ahd  by  the  faid  poyntes  pro¬ 
duce  a  right  li  ne,  it  (hall  cube  the  lyne  E  G  beyng 
produced  in  the  pointe  N,  w hich  ('ball  be  the  cen¬ 
tre  of  the  circle  by  the  fame  Corollary  of  the  firft 
of  this  booke,if  you  imagine  the  right  line  C  D  to 
be  drawen  and  to  be  deuided  perpendicularly  by 
the  lyne  L  M,  which  it  muft  needes  be  as  wc  hauc 
before  proned.  And  here  note  that  to  do  this  me¬ 
chanically  not  regardyng  demonftration  ,  you 
neede  onely  to  markethe  poyntes  where  the  cir¬ 
cles  cut  the  one  the  other,namely,  the  poyntes  E, 

G,andL,M,  and  by  thefe  poyntes  to  produce  the 
lines  ii  G  and  L  M  till  they  cut  the  one  the  other, 
and.  where  they  cut  the  one  the  other,  there  is  the 
centre  of  the  circlets  you  fee  herein  the  feconde 
figure. 


"Pd.ij' 


*A  ready  way 

to  finde  ohs 
the  center  of  4 
circle  co  mmo* 
ly  yjed  a- 
mmgft  artb- 
feen. 


ConjlmUm. 

Demonftm- 
nm » 


23.  'Theoremc*  1  he  16*  Tropofition* 

Equall  angles  in  e quail  circles  eonfifl  in  equall  cirmferences * 
whether  the  angles  be  drawen  from  the  centre  syor  from  the 
circumferences . 

^ppofe  that  thefe  circles  A  3  C  and  ID  E  Ffe  equall.  And  from  Mr 
\centres  piamely 3the pointes  G  and  If  let  there  he  drawen  thefe  equal! 
'angles  3G  C  and  ELLF:  andhkewife from  their  circumferences  thefe 
equall  angles  3  AC  and  EDF.  Then  1 fay3  that  the  circumference  3  If  C 
is  equall  to  the  circumference  E  LF .  Draw  (by  the firft petition)  right  lines 
from  3  to  Cj  and from  E  to  F.  And for  afmuch  as  the  circles  A3  C  and  DEE 
are  equally  he  right  lines  aljo  drawen  front  their  centres  to  their  circumferences  „ 
are  (by  the firft  definition  of  the  third  )  equall  the  one  to  the  other  .Wherefore 
thefe  two  lines  3  G  and 
G  C3are  equall  to  thefe  two 
lines  E  FI  and  EL  F.  And 
the  angle  at  the poynt  G  is 
equall  to  the  an  fie  at  the 
point  LI:  Wherfore  ( by  the 
4  *of the  firfijt  he  hafe  3  C 
isequaMtoybafe  EE.  And 
for  afmuch  as  the  angle  at 
thepbynt  A  is  equall  to  the  angle  at  the  point  D}  therefore  the  fegment  3, AC 
islilye  iq  ihe  fegment  E  D  F.  And  they  are  defcribed  W>pon  equall  right  lines  3  C 
and  E  F .  3ut  like  fegmentes  of  circles  defcribed  ypon  equall  right  lines }  are  (by 
the  24.  of  the  third)  equall  the  one  to  the  other ,  Wherefore  the  fegment  3  A  C 
is  equall  to  the fegment  EDF.  And  the  ‘whole  circle  A3C  is  equall  toy  'wide 
circle  DEE .  Wherefore  ( hy  the  third  common J  en  tence)  the  circumference  re* 
mayning  3  JfiC  is  equall  to  the  circumference:  remayning  ELF  .  Wherefore 
equall  angles  in  equall  circles  confifi  in  equall  circumferences whether  the  angles 
be  drawen  from  the  centres  or  from  the  circumferences:  'which  Seas  required  to  he 
demonstrated. 

....  "■  • 

\  -  \  :  V 

:  .  '  - 

he  %^Theonfae*  .  ■  -*7-  'Trop.ojttion. 

In  equall  circles  the  angles  which  conffjn  equall  .circumfer 
rences 9ar e  equal!  the  one  to  the  other  ?  whether  the  angles  he 
drawen  from  the centres yor from  the  circumferences « 

■Suppofe 


*  i 


cfSuclidesElementcs.  FoLxou 

\  yppofij  Cfoefe  circles  AB  C,and  ID  E  F,  be  e quail.  Andy>pon  thefe 
equall  circumferences  of  the fame  circles yiamely  gbpon  B  C  and  EF, 

I  let  there  con  ft f  thefe  angles  IB  GC  and EH  F  drawen from  the  cen - 
tres  yindalfo  thefe  angles  BAC  and  ED  F  drawen  from  the  cir* 
cumferences .  T  hen  I fay,  that  the  angle  BGC  is  equall  to  the  angle  EHF, 
and  the  angle  BAC  to  the  angle  ED  F.  If  the  angle  B  G  C  be  equall  to  the  an * 
gle  E  H  Fgthen  it  is  manifelljhat  the  angle  BAC  is  equall  to  y  angle  ED  F 
(by  the  20.  of  the  third).But  if  the  angle  B  G  C  be  not  equall  toy  angle  EHF 
then  is  the  one  of  them  greater  then  the  other .  Let  the  angle  BGC  be  greater 
And  (  by  the 
21'ofthefirft) 
ypon  the  right 
line  BGPand 
ynto  the  point ; 

geuen  in  it  G,  c?  \  . 

make Ipnto  the ,  /  \ 

angle  EHF  \  I  j  \vY 
an  equall  am  V  J  /  \\Y 

gurnet  xy  \A 

(by  the  26*  of  b"^~ — — 

j  third') equall  \ 

angles  in  equall  circles  confifl  fyo  equall  circumferences  whether  they  be  drnwm 
from  the  centres  or from  the  circumferences .  Wherefore  the  circumference  B  ^ 
is  equall  to  the  circumference  EF .  Bui  the  circumference  EF  is  equall  to  the 
circumference  B  C:  Wherefore  the  circumference  B  F(  alfo  is  equall  to  the  cir* 
eumference  BC ,  the  lejfe  to  the  greater :  Hfhichis  impofiihle ,  Wherfore  the  an* 
gle  BGC  is  not  Unequal!  to  the  angle  E  HE:  Wherefore  it  is  equall.  And  (by 
the  20.  of the  third )tbe  angle  at  the  point  A  is  the  halfe  of the  angle  B  G  C:and 
(by  the fame )  the  angle  at  the  point  D  is  the  halfe  of the  angle  EHF.  Where * 
fore  the  angle  at  the  point  A  is  equall  to  the  angle  at  the  point  D.  Wherefore  in 
equall  circles ,t he  angles  ^ohich  confifl  in  equall  circumferences  yare  equall  the  one 
to  the  other  pxhether  the  angles  be  drawen  from  the  centres  or  from  thecircumfe * 
rences :  Ichich  leas  required  to  be proued. 


Demnftr&* 

tioh  leading 
to  an  impof- 
fihlitko 


$&The  25.  'Thecremc.  Ifhe  28.  Tropofitlon. 

In  equall  circles,  equall  right  lines  do  cut  assay  equall  cir* 
cumferences  ^ he  greater  equall  to  the greater, and  the  lefie  e* 
quad  to  the  le [te* 

.  S)d.ij. 


Suppofc 


Cwflruftm. 


DemnBra* 

sim,. 


The  tonncrfe 
of  the  former 
Fropofition. 

Conflru  ction • 

T>emon$rA~ 

tmi. 


F^pofe tfatthefid'rchs  ABC*,  dndfD  M-Ffic efiimll.  Andfiiftkefrf hi 
J  there  be  graven  fhefifyuaH  tight  iikexfB  C and  EF}  Tthichfit  cytawpy 
llthefe  dHtmfenntei  B  A  C ahfidbE  F  beirig  ihegrmter  }<Malfo  fhefo 


titcimference  B&  V  is  eqmfy  tithe  lejfe  circumference  E  HF.  Take  (by  the 
fifftof the  third)  the  centres  dflhedYctCsfind  let  the  fume  he  the  pointts  Efand 
<L .  And  dfttw  thefe  fight  fines ?  3fB}  Jf€>  and  L  F .  And  for  afmpchps 

the  drcltsrane^diyMrfit^hj  thefirft  definition  of  the  third)  the  fines  "pBch 
are  dycnvenfro  thFcenM'  -  ■  "■  .  .  ^ 

tres  are  eqtcall  .Where* 
fore  thefe  two  lines  (B  If 
mid  JfC y  are  equall  to  /  jg. 

thefe, two  lines*  E  E  and  \  p 
E  f.  And  (by  fuppofiti* 
onjthe  bafe  EC  is  equall  c, 

to  the  ba(W 
fo 


•My  fir  I 


the  angle  B'ffC'i's  equall  to  the  angle  ELF.  (But  (by  the  26*  of  the  third) 
equal!  angles  draw en from  the  centres }conffl  tpon  equall  circumferences, Wher* 
fore  the  circumference  BGCis  equall  ti  the  circumference  E  H F;  andj  nhok 
-Wflt  fiJMGAs  equall  to  the  fihole  circle  STEF  .‘wherefore  the  circumference 
yemqyningB^G,  is(by  the  third  common  fentenee)  equall to  the drcumfmnct 
rmdynirtg  ESTF.  Wherefore  in  circles  (equall  right  lin^s  do  cut  away  email 
cfcufifirtncesythegreater  equall  to  the  greater ^and  thekffe  equall  to  the  ieffi: 
fihichipasrequdredhbeproued.  -'"w"  ‘  ,  "  "...  • 

■  .>  '  ‘  ■  •  •  •  ■  - 

i&tfhepretoe.  ,  J 

■  f  %  1  v  ^  ^  :*'t "  fi/  ■ 

M  ^UAlickcks finder  ^uaUfircumfir^cfis  ^efukended 


enm-mig 


hefe  circles  A  BC  and  S)  EF fie  equall,  ytnd in  them 
let  there  fie  tafjm  *  thefe  equall  circumferences* 3  B  G  C  and  EM  Ft 
anddrawe  thefe  right  lines  (B  C  ana  EFfThenlfyythat  the  right 
BC  is  equall  to  the  right  line  E  F .  T dke  ( by  thefirfi  of)  third ) 
tlk  ’Centres  of  the  circles  y  anil  let  them  be  thepointes  Jf  and  E,  and  draw  thefe 
rigbtiines  If  Bj  IfC,  E  EyE  F.-, find for dfrhucb  as  the  circuiFferenceB  G  C 
is  equall  ifitlfi  circumference  EH  Fyhe;. angle B:hf 6  is  equall  tofiangle  ELF 
( by  the  27,  of  the  third)  .  And  for  afmucb  as  the  circles  jC BC  and  Sj  E  F are 
equalltheone  to  the  Miter yherefire  (  by  the firfi  definitwnofitbe  thirdfme  lines 
'  *  ’  Tehich 


■  jVi  *  1 


a  i 


FoIaoz* 


be  cen  t  r  es 
are  equall. 

WhWtfbri- 

z*'he(ei 


K-fy 


are 


nw, 

t  i-T  *V'.» 


tpejen jjges 
jf  $.  and 

fir  jtf  they  ^  .vi  ,  .  f,  v  , 

comprehend  equal! angles .  Wherefore  (lip  the  4  ♦  dfihefrftjthe'hnfe  !BC 
quail  to  thebafe  EF  .Wherefore  in  equall  circles  "under  equdfl  Circulnferekce't^ 
are  fuhtended  equal l  right  lines :  yhich  yds  required  to  he  demon frated. 


The  4..'  Troblenie .  The  30.  Tfopoftim, 

'  a  circumferencegeuen  into  Wo  eqtialtparUW 

1  1 1  is  reauired  ttfdi* 


W^yp^Fppofe  that  the  circumference geuen  hejiHb3. 
%ide  the:?ircumfererice..A0  %ihib  two  equall p, 


artes.  (Draw  ar  is 


tWfoMiVr'dyfe^fp  ZtfitoA'B  a  perpendicular  Unit  &  And  draw thefe right 
Him  A  3D  Md0B:Andfbrafmuch  as  the  line  A  €  is  equall  to  the  ImeCBfsp 
the  line  C  0  is  common  to  them  both y  there* 
fore  thefe  two  lines  A C  and  C(D  are  equal t 
to  thefe  two  Ims  rB  C  mdC(D.And( by  the 
4. petition,)  the.  angle  A  CD  is  equal!  to  the 


'  cV  ’  ds.  *  -  •  -  ;  7  V  X  S 

the,  bafeA  0  is  effaff.  to  the  b.afe  D  T$f  Bui 
equall  right  lines  do  cut  away  equall  eircum? 
ferefcespthe greater  equall  to  the  greater ftP 


"  a 


zjgliheoreme.  ^Fb.e^u 

fnaemk  m thefommrpk  &  a.rigfamgfr. 

*&d]wi*  bill 


Conftmtm . 


Dmonftr <#- 
tm r. 


The  firft  part 
nfthtt  ThcQ • 

reme. 


Smn&pwt* 


"lUtip&rt. 


fTbe  third fBoo^e 

but  an  angle  made  in  the  fegment greater  then  the  femieirck 
islejfethena  right  angle ,and an  angle  made  in  the  fegment 
lejje  then  the  femicircle,  is  greater  then  aright  angle .  Jnd 
moreouer  the  angle  of  the  greater  figment  u  greater  then  a 
right  angle:  and the  angle  of  the  lejje fegment  is  lefe  then  a 
rightanglc.  T 

K|||  P°fi  ^dt  the  cFcle  be  AB  C  D,and  let  the  dimetient  of  the  circle  be 

right  line  BC,  and  thecetre  therof the  point  R.  And  take  inthefr 
miclrcle  a  point  at  all  auentpres ,and  lei  the  fame  he  (D*  And  draw  theft 
right  lines  B  A, A  C^A Dyxndfp  C.Then  I 
Jay  that  the  angle  in  the  femicircle  BAC, 
namely,  the  angle  B  AC  is  a  right  angle. 

And  the  angle  A  B  Clinch  is  in  the  fegment 
AB  C  being greater  then  the  J'emicircle ,  is 
lefe  then  a  right  angle.  And  the  angle  A  ID 
C -ftbicb  is  in  the  fegment  A  ID  C  being  lefe 
then  the  femicircle  is  greater  the  a  right  an « 
gle.-Draw  a  line  from  the  point  A  to  the  point 
E,and  extend  the  line  B  A  Imto  the  point  K 
And  forafmuch  as  the  line  BE  is  equall  to 
the  line  E  A,  ( for  they  are  dr awen  from  the 
centre  to  the  circumference)  therfore  the  an * 
gleEAB  is  equall  to  the  angle  E  B  A  (by 
the  q.of  the firH).  Againe forafmuch  as  the 
line  A  E  is  equall  to  the  line  E  C,  the  angle 

ACE  is  ( by  the fame)equall  to  the  angle  C  A  E.Wherfore  the  ^ hole  angle1. B  A 
C  is  equal!  to  thefe  two  angles  ABC  and  A  C  B.But  the  angle  FAC  lobich  is 
an  outward  angle  of  the  triangle  AB  C  is  (by  the  32.  o  f  the  firH)  equall  to  the 
two  angles  ABCzy  AC  B. Wherfore  the  angle  B  AC  is  equall  to  the  angle  F* 
A  C. Wherfore  either  of  them  is  a  right  angle .  Wherfore  the  angle  BAC  t>bich 
is  in  the femicircle  BAC  is  a  right  angle. 

And forafmuch  as  (by  the  17*  of  the  firH)  the  two  'angles  of the  triangle  A 
'B  C, namely  ,A  B  C  and  BAC  are  lefle  then  tworight  angles ,  and  the  angle  B 
'A  C  is  a  right  angle. 'Therfore  the  angle  AB  Cis  lefe  then  aright  angle, and  it 
is  in  the  fegment  ABC  lohich  is  greater  then  the Jemkirde . 

And forafmuch  as  in  the  circle  there  is  a  figure  offourefides ,  namely,  AB 
C  D.  But  if whin  a  circle  be  defcribed  a  figure  offoure fides ,  the  angles  therof 
•tyhich  are  oppofite  the  one  to  the  other  are  equall  to  two  right  angles  (by  the  22» 
*&f the  third)  Wherfore  (by  the  fame )  the  angles  ABC  and  ADC  are  equal l 

.  te 


ofSucliJes  Ekmentes .  FoLio^ 

to  two  right  angles  SB  ut  the  angle  AB  C  is  lefie  then  a  right  angle. Wherfore  the 
angle  remaining  AD  C is greater  then  a  right  angle ,  and  it  is  m  a  figment 
'W Inch  is  kfie  then  the  femicirck \ 

Now  ctlfi  I  jay  that  the  angle  of  the  greater  figment ,  namely  ,  the  angle 
yghicb  is  comprehended  'tinder  the  circumference  A  B  C  and  the  right  line  A  C 
is greater  then  aright  angle , and  the  angle  of  the  lefie  figment  comprehended 
tinder  the  circumference  AID  C ,  and  the 
right  line  AC  is  kfie  the  aright  angle; thick  / 

may  thus  he  proued.  Forafmuch  as  the  angle 
comprehended  lander  the  right  lines  BA  and 
AC  is  aright  angle,  therfore  the  angle  com * 
prehended -louder  the  circumference  ABC 
and  the  right  line  A  C  is  greater  then  a  right 
angle :  for  the  ~^bole  is  euer greater  then  his 
part  ( by  the  p.  common Jentence. 

Againe for af much  as  the  angle  comprt * 
bended  louder  the  right  lines  A  C  and  A  F 
is  a  right  angle ,  therfore  the  angle  com * 
prehended  lender  the  right  line  C  A  and  the 
circumference  ADC  is  lefie  then  a  right  an* 
gle. Wherfore  in  a  circle  an  angle  made  in  the 
femicirck  is  aright  angle, but  an  angle  made 
in  the figment greater  then  the femicircle  is  lefie  then  a  right  angle, and  an  an* 
gle  made  in  the fegment  lefie  then  the femicirck,  is  greater  then  a  right  angle < 
And  moreouer  the  angle  of the  greater figment  is  greater  then  a  right  angle:  ejr 
the  angle  of  the  lefie  figment  is  lefie  then  a  right  angle  :  lohich  mas  required  to 
be  demonstrated.  -A 

_  M  other  demonstration  to  proue  that  the  angle  B  AC  is  a  right  angle. For* 
dfmuch  as  the  angle  AFC  is  double  to  the  angle  B  A  E(by  the  ^*cf thefirfi ) 
for  it  is  equal!  to  the  two  inward  angles  mhich  are  oppofite.  But  the  inwarde  an* 
gks  are  ( by  the  5.  of the  fir  A)  equall  the  one  to  the  other, and  the  angle  A  E  B  is 
double  to  the  angle  E  A  C.  Wherfore  the  angles  A  EB  and  A  EC  are  double  to 
t  he  angle  B  A  C.  But  the  angles  A  E  B  and  A  EC  are  equall  to  two  right  an* 
gks:  Wherfore  the  angle  B  AC  is  a  right  angle.  Which  mas  required  to  he  de* 
i  monSirated. 

y^Correlarj, 

fb,  •  .  ,  ‘  »>  -  ‘ 

Her  eby  11  is  niantfjl fib  At  if  in  a  triangle  one  tingle,  he  etfuall 
to  the  two  01  her  angles  remaining  the  fame  angle  is  a  right 


The  fourth 

part. 


The  eft 
USt  part* 


•Another  De¬ 
monstration 
to  prone  that 
theang’e  in  a 
femicircle  is  4 
right  angle. 


A  CtmUasy. 


A*t  addition 
•■affelifatiHS, 


\ Denton  Sira - 1 
tion  leading 
So  an  abjurdi* 

th i. 


iAn  addition 
of  Cam  fane. 


X  . 


angle', for  that  the  fide  angle  to  that  one  angle  ( namely ,  the 
angle  which  is  made  of  the fide  produced  without  the  trian~ 
gle)  is  e  quail  to  the  fame  angles Jaut when  the  fide  angles  are 
equal!  the  one  to  the  other ft hey  are  alfo  right  angles . 


f  yfn  addition  off’elitarius. 


v\  V 


ppofite  vnto,  the  right 

'  ‘  C  ‘  '  '  \\  V  :  '*.?  '•  i  .' 

:\.  ■  m*  •  'c;  . :  ■  v  v 

1  \  .  '  • A  i  A.:  •  ~v  Y>  v  Y 


If  in  a  circle  be  infcribed  a  redangle  triangle  ,  the  fide  o 
angle  fhall  be  the  diameter  of  the  circle. 

Suppofe  that  in  the  circle  ABC  be  infcribed  a 
re&anglc  triangle  A  B  C,  whofe  angle  at  the  point 
B  let  be  a  right  angle. Then  I  fay  .that  the  iide  A  C 
is  the  diameter  of  the  circle .  -For  if  not,  then  fhall 
the  centre  be  without  the  line  A  C,as  in  the  point 
E.And  draw  a  line  from  the  poynt  A  to  the  point 
E,& produce  it  to  the  circumference  to  the  point 
D ;  and  let  A  E  D  be  the  diameter  :  and  draw  a  line 
from  the  point  B  to  the  point  D.Now(by  this' 3  1*  Aj 
Propofitio)  the  angle  A  B  D  flia.ll  be  a  right  angle, 
and  therefore  fhall  be  equall  to  the  right  angle 
ABC, namely,  the  part  to  the  whole  :  which  is  ab- 
furde.  Euen  fo  may  we  proue,  that  the  centre  is  in 
no  other  where  but  in  the  line  A  C.  Wherfore  A  C 
is  the  diameter  of  the  circle  :  which  was  required 
£0  beproued. 

f  Jin  addition  ofCampane. 

-  -  '.v 

By  thys  31.  Propofition,and  by  the  16.  Propofition  of  thys  booke,  it  is  mani¬ 
fold, that  although  in  mixt  angles, which  are  contayned  vnder  a  right  line  and  the 
.circumference  of  a  circle,there  may  be  geuen  an  angle  Idle  &  greater  then  a  right 
angle, yet  can  there  neuer  be  gene  an  angle  equall  to  a  right  angle.For  euery  fe&i- 
on  of  a  circle  is  ey  ther  a  femicircle,  or  greater  then  a  femicircle, or  Idle, but  the  an¬ 
gle  of  a  femicircle  is  by  the  i<5.of  thys  booke,  Idle  then  a  right  angle,  and  fo  alfo  is 
the  angle  ofa  Idle  Fedion  by  thys  3 1  .Propofition :  Likewife  the  angleofa  greater 
jedfion,  is  greater  then  a  right  angle,  as  it  hath  in  thys  Propofition  bene  proued. 

he  28.  Theorem,  The  3 zfPropoftion . 

If  a  right  line  touch  a  circle ^and from  the  touch  be  draymen  a 

nght  line  cutting  the  circle:  the  angles  which  that  line  and 

the  touch  line  maM,are  equall  to  the  angles  which  confjl  in 

the  alternate  fegmentes  efthe  circle . 

Y  Vppofe  that  the  right  line  EF  do  touch  the  circle  jlECD  in  the 
\  point  F> ;  and  from  the  point  F>  let  there  he  drawen  into  the  circle 
\AFCD  a  right  line  cutting  the  circle ,  and  let  the  famehe  FED. 
"  T hen  I  fay ?  that  the  angles  Svhich  the  line  D  together  Ipith  the 

touch 


of  Smiths  Ekmefrte's.  Pol .  104,* 

touch  line  EF  do  make ,  are  equallto  the  angles  "Which  are  in  the  alternate  peg* 
mentes  of  the  circle  ythat  is  y  the  angle  FBD  is  equall  to  the  angle  "Which  conji* 
fieth  in  the  fegment  B  A  Dy  and  the  angle  EBD  is  equall  to  the  angle  scinch 
conflict!)  in  the fegment  BCD.  f aije  l)p  (by  the  it,  of the  frJl)fromy  point 
B  » the  right  line  EF  a  perpendicular  line  B  A .  And  in  the  circumference 
B  D  take  a  point  at  all  aduenturcs}and  let  the fame  he  C.  And  draw  thefe  right 
lines  ADyD  C,md  CB.  And  for  afmuch  as  a  certaine  right  line  EF  tou* 
cheth  the  circle  A  B  C  in  the  point  B>  and from  the  point  B  ‘inhere  the  touch 
is  ra^fedio^nioihe'iouch  Im&perpen*  ■ 
dicular  B  A.  fherfore(hy  the  is?«  of  the 
third )  in  the  line  B  A  is  the  centre  of  the 
circle  ABC  D.Wherforey  angle  A  D  B 
being  in  the  femictrcle,  is(by  the  Z1*  of  the 
third )  a  right  angle  .  Wherefore  the  an* 
gles remaining  BAD  and  A B Dyare 
equall  to  one  right  angle .  But  the  angle 
AB  Fisa  right  angle.Wherefore  the  an* 
gle  ABF  is  equall  to  the  angles  BAD 
and  A  BD .  F  a  he  away y  angle  ABD 
* which  is  common  to  them  both.Wherefore 
the  angle  remayning  DBFfs  equall  to 
the  angle  remayning  BAD }  * which  is  in 
the  alteimate fegment  of the  circle .  And  for  afmuch  as  in  the  circle  is  a  fgure  of 
power fades  ymnelyyA  3  C  D)therfore(by  the  22*  of  the  third)  the  angles  "Which 
are  oppofite  the  one  to  the  other yare  equall  to  two  right  angles.  JVberforethe  an* 
gles  B  A  D  and  BC  Dy  are  equall  to  two  right. angles .  But  the  angles  DBF 
and  D  B  E}  are  alfo  equall  to  two  right  angles .  Wherefore  the  angles  DBF 
and  DBE^are  equall  to  the  angles  BAD  and  BCD.  Of  "which  "We  bant 
pmied  that  th; angle  BAD  is  equall  to  the  dngle  D  BF .Wherefore  the  an* 
gleremaynmg  D  B  Ey  is  equall  to  the  angle  remayning  DCB}  "which  is  in  the 
alternate  fegment  of  the  circle-  ytamelyjn  the fegment  D  C  B.Ifther fore  aright 
line  touch  a  circle yand  from  the  touch  be  drnwen  a  right  line  cutting  the  circle: 
the  angles rwhich. that  line  and  the  touch  line  make y  are  equall  toy  angles  "Which 
cmfft  my  alternate fegmentes  of  the  circle :  * which  - Was  required  to  be  proued. 

In  thysPropoimon  may  be  two  cafes .  For  the  line  drawer*  from  the  touch  and 
Siting  the  circle,,  may  ey  they  patfe  by  the  centre  or  not .  If  it  pafle  by  the  centre, 
then  is  it  manifeft  (by  the  i8  .  6fthysbooke)  that  itfalleth  perpendicularly  vpon 
the?  touch  line,and  deuideth  the  circle  into  two  equal!  partes,  fo  that  all  the  angles 
itf  &he  feinidrcle,are  by  the  fbrihef  Propofitioh, right  angles,  and  therfore  equal! 
to  the  alternate  angles  made  by  thcfayd  perpendicular  line  and  the  touch  linc.Ific 
paiTe  not  by  the  centre,  then  followe  the  conftm£tion  and  dcmon&adon  be¬ 
fore  put.  '' 


Cenjbufiiots, 


Bmortfira- 

tion. 


Two  cafes  in 
this  Prope/i* 
tisn. 


Three  cafes  in 
this  tiopo/i- 
tion. 

The  firtt  (tfe, 
Conftruftion. 


Vemon/ka- 

tioti. 


Thefecond 

safe. 


l&The  ^.Trobleme.  "The  fifProfoJitm. 

t,  v  V*  .  •  -  '•  •  r  '•  rj  •  i  .  .«  v <? 

Vppon  a  right  lynegeuen  to  defcnbe  a  fegment  of  a  circle » 
mickfhall  contaynean  angle  e quail to  a  reBilirie  angle geue. 

$rppofe  that  the  right  linegeuen  he  A  Bqand  let  the  reBiline  angle gc* 
iten  be  C.It  is  required  Tpon  the  right  line  geue  A  Bto  defiribe  a [eg* 
ment  of  a  circle  "Which Jhall  contayne  an  angle  equal!  to  the  angle  C 
-Now  the  angle  C  is  either  an  acute  angle ?  or  a  right  angley  ordn  ob * 
tufe  angle* 

Firfty  let  it  be  an  acute  angle  as  v  ■ 

in  the firft  defcription.A n d( hy  the  23 
of  the  firfi)  Wpon  the  right  line  A  B 
and  to  the  point  in  it  A  defcnbe  an 
angle  equal  to  the  angle  Cy  and  let  the 
fame  be  (DAB*  Jf her fore  the  angle 
ID  Addis  an  acute  angle .  From I the 
point  A  raife  T>p(by  then •  ofyfrfr) 

''mi  to  the  fine  AD  a  perpendiculer 
line  A  FL  Andfhy  the  J  9  •  of  the  frjl ) 
deidde  the  line  A  B  into  two  equall 
partes  in  the  point  F.  And  (by  the  n. 

'of  the  Jhfiefftofn  ihepbintFfaife&p  Tmto  the  line  A  IB  a  perpendicular  tynq 
FGfnddYinn  dime fromGtoB.Andforafmdchas  the  line  A  F  is  equall  to  the 
line  F  3  and  the  line  FG  is  comm  on  to  them  both y  therfore  thefe-  two  lines  AF 
'and  FG  are  equall  to  thefe  two  lines  F  B  and  F  G:  and  the  angle  AFG  is  (by 
the  4g)Ctidon)  equall  to  the  angleG  F  B.  JVherfore(hy  the  4.  of  the  fame )  the 
bafe  A  G  is: equall  to  the  bafe  G  B.  Wherfore  making  the  centre  G  and  thefpace 
GA  dejdrtbe  {by  the  5  .peticion)  acircle  and  it  fall  pajfe  hy  the  point  B :  de* 
fcribefuch  acircle  &  let  the  fame  be  A  B  Be  Arid -draw  'a  line  from  E  to  B.Nop. 
fora'/imch  'as  from  the  ende  of  the  diameter  A  E,  namely }  from  the  point  A  is 
Urawen  a  right  line  A  D  making  together  "With  the  right  line .  A  E  a  right  am 
gkfherfrr/fhythe  cor fellary vf the  16.  of  the  third)  the  line  A  D  toucheth  the 
circle  'A*  B"E.  AndfotafMmhdsAcertaint  "'right  line  AD.  toucheth  the  circle 
A1. B  Egs  from  the  point  A  Cohere  the  touch  is  (is  drawen  intoy  circle  a  certaine 
right  line  A3:  therforefhy  the  32.  of the  third)  the  angle  D  A  Bis  equal!  to  the 
angle  A  EByvhich  is  in  the  dltefndte fegment  of  the cir  cle. But the angle  DAbB^ 
is  equall  to  the  angle  C ^herfore  f  he  angle  C  is  equall  to  the  angle  A  E  B.IVher* 
fore  hpon  the  right  linegeuen  A  Bis  deferibed a  fegment  of  a  circle  "which  com. 
tayneththe  angle  AEBfWhkh  is  equall  to  the  angle geuen /tamely yto  C.  : 

Butmufi'ppofe  that  the  angle  Cbe  a  right  angle.  It  is  againe  required 

■pm  *** 


of Sticlides  Elements s.  FoL 

port  the  right  line  ABto  defcribe  a  feg* 
merit  of  a  circle ,  which  JhaU  contayne  an 
angle  equal  to  the  right  angle  C. Defcribe 
againe  "Upon  the  right  line  A  B  and  to  the 
point  in  it  A  an  angle  BAD  equal  to  the 
reclilme  angle  geuen  C  (by  the  23*  of  the 
fr$l)  as  it  is  jet  forth  in  the  fecond  de*  ^ 

J crip t ion.  And  (by  the  10 ,of  the  fir  ft )  de *  ' 
nide  the  line  AB  into  two  equall  partes 
in  the  point  F.  And  making  the  centre  the 
point  F  and  the J 'pace  F A  or  FB  defcribe 
(by  l he  3 . petition)}  circle  ABB.  Wher*  ** 
fore  the  right  line  A  D  toilcheth  the  cir* 

cle  A  FB  :for  that  the  angle  BAD  is  a  right  angle W her fore y  angle  BAD 
is  equall  to  the  angle  "Which  is  in  thefegment  A  E  B}for  ^Je  angk  ‘Which  is  in  a 
Jemicircle  is  a  right  angle(by  the  31*  of  the  third )  But  the  angle  BAD  is  equal 
to  the  angle  C.  JVherfore  t here  is  againe  defcribe  d lapon  the  Itne  AB  a  Jegment 
of a  circle  jUamely  }A  FBpwhicb  contained?  an  angle  equall  to  the  angle  geuen 
namely  }to  C. 

But  now  fuppofe  that  the  angle  C  be  an  obtufe  angle.  Vpon  the  right  tine  AB 
and  to  the  point  in  it  A  defcribe  (by  the  of  the firft)  an  angle  BAD  equall 

to  the  angle  C:  as  it  is  in  the  third  defcription.  And from  the  point  Arayjb  Wp 
Tmto  the  line  AD  a  perpendiculer  line  A  B 
(by  then-  of the frjl )  And  agayne  by  the 
10  .of the fir  ft)  deuide  the  line  A  B  into  two 
equall  partes  in  the  point  F.  And  from  the 
point  F  rayfc  lop  Imto  the  line  A  Ba  per  pi* 
dicular  line  F  G  (by  the  11.  of  the  fame) 
draws  a  line  from  G  to  B.  And  now  forafi 
much  as  the  line  A  F is  equal  to  the  line  FB} 
and  the  line  F G  is  common  to  them  both - 
th  erf  ore  thefe  two  lines  A  F-aiidFG  are  e* 
quail  to  thefe  two  lines  B  F  and F  G :  and 
the  angle  A FG  is  (by  the  4  ♦  peticion)  equall  to  the  angle  B  FG :  "wherfore  (by 
the  4  *  of  the  fame)  the  bafe  AG  is  equall  to  the  bafe  G  B.Wherfore  making  the 
centre  G/ind  t he f pace  G  A defer ibe( by  the  3* peticion)a  circle  and  it  fhall pafie 
by  the  point  B:  let  it  be  deferibed  as  the  circle  AEB  is.  Andforafmuch  as  from 
the  ende  of the  diameter  A  B  is  drawen  a  perpendiculer  line  AD  therefore  (by 
the  correllary  of the  16.  of the  third)  the  line  AD  touche  th  the  circle  AEByr 
from  the  point  of  the  touche jnamely,  A  js  extended  the  line  A  B. JVherfore  (by 
the  32‘  of the  third )  the  angle  BAD  is  equall  to  the  angle  A  HB  lohich  is  in 
the  alternate  fegment  of  the  circle. But  the  angle  BAD  is  equall  to  the  angle  C 

Ee.j.  Wher <* 


E 


Demnffrs* 


Thethiri 

cafe., 

CmfmUwh 


Qmenjlrfa 
mm » 


Cotijlruftion . 


Vemonjlra - 
tmu 


Wherefore  the  angle  Tohich  is  in  the  fegment  A  HE  is  equal!  to  the  angle  C. 
Wherfore  ypon  the  right  linegeuen  A  Ejs  defcribed  a fegment  of  a  circle  AH 
Epvihich  contayneth  an  angle  equal!  to  the  angle gotten ,  namely  .  C:  Tsthich  leas 
required  to  he  done.  \  .  4 

S^fThe  C.Troblems*  The  54.  . £ Tropofition . 

From  a  circle geuen  to  cut  away  afeBion  which  fhal  contains 
an  angle  e  quail  to  a  reBtline  angle  geuen* 


j{:  Vppofe  that  the  circle  geuen  he  AC  and  let  the  reHiline  angle  geuen 
^y^r.be  D.  It  is  required  fro  the  circle  A  3  C  to  cut  away a fegment  "Which 
fall  contayne  an  angle-  equal!  to  the  angle  D.  Draw(bythe  17  of  the 
/  third)  a  line  touching  the  circle,  and  let  the  fame  be  EE:  and  let  it 

touche  in  the  point  3.  And  (by  the  23.  of 
the  prfi)  hpon  the  right  line  EF  and  to 
the  point  in  it  3  defer ibe  the  angle  F  EC 
equal l  to  the  angle  ID.  ISiom  for aj much  as 
a  ccrtayne  right  line  E  F touche th  the  cir* 
cle  A3  C  in  the  point  3:  and  foray  point 
of  the  touche ynamely fB }  is  drawn  into  the 
circle  a  ckrtaine  right  line  3  C ,  therefore 
(by  the  32*  of the  third)the  angle  FBC  is 
equal!  to  the  angle  3  yfC  Tvhich  is  in  the 
alternate  fegment.  But  the  angle  F 3  C  is  E 

equal!  to  the  angle  D.  Wherfore  the  angle 

3  AC  Tehich  con  file  th  in  the  fegment  3  A  C  is  equal 1  to  the  angle  55.  Where* 
fore from  the  circle  geuen  A  3  C  is  cut  away  a fegment  Eg!  C gtobich  containetb 
an  angle  equal 1  to  the  reHiline  angle  geuen:  Hitch  Teas  required  to  be  done . 

\  ^  a*  —  j>  )  v  \  '  1  '  •  -)  ■  ' 

T* he  zp .  Theorems ,  The  ffropojition • 

If  in  acircle  two  righ  times  do  cut  the  one  the  other Jtherect* 
angle paraUelograme  comprehended  vnder  the  fegmentes  or 
parts  of  the  one  line  is  equaU  to  the  reBangle  paraUelograme 
l  comprehended  vnder  the fegment  or  partes  of  the  other  line . 

'■  '  V-  .  ;  :  .  .  ,,  '  .  ,, 

«  I:  .  ",  -  -  ; .»  '»  ,  y  ;  \\  '  •.  v  '*  -  "4- 

Etthe  circle  be  A E  C  Dy  and  in  it  letthefe  two  right  lines  AC  and 
E  D  cut  the  one  the  other  in  thepoint  E.Thenlfay  that  the  reHangh 
farallelogramme  contayned  lender  the  partes  A  E  and  EC  is  equal!  to 

the 


ofSuclitles  Elementes ,  Fq/aq6, 


be  drawen  by  the  ce  ntre /hen  is  it  manifeji  fhat for 
as  much  as  the  lines  A E  arid  E  C  are  e quail  to  the 
lines  ©  E  and  E  B  by  the  definition  of  a  circle }  the 
re  cl  angle  parable  lograme  alfo  contayned  Imder  the 
lines  A  E  and  E  C  is  equall  toy  reft  angle  paralle * 
lograme  contained  bnder  the  lines  IDE  and  ED. 

But  now fippofe  that  the  lines  A  C  and  ID  B  be 
not  extended  by  the  centre ymd  take(by  the  i.  of  the  third)  the  centre  of  the  cir* 
cle  AD  C  D}and  let  the fame  be  the  point  F,  and  from  the  point  F  draw  to  the 
light  lines  AC  and  ©  B perpendicular  lines  FGand  F  H.  (by  the  n,  of  the 
fir  ft)  and  draw  thefie  right  lines  F  BfF  C  pnd  FE. 

Andforafmuch  as  a  certaine  right  line  FG  drawen 
by  the  centre yutteth  a  certaine  right  line  AC  not 
drawen  by  the  centre  in  fuel?  forte  that  it  rnaketh 
right  angles }  it  therfore  deuideth  the  line  A into 
two  equall  partes  (by  the  3.  of the  third).  Wherfore 
the  line  AG  is  equall  to  the  line  GC.  And  foraf 
much  as  the  right  line  AC  is  deuided  into  two  e* 
quail  partes  in  the  point  G}  and  into  two  Unequal! 
partes  in  the  point  E:  therfore  ( by  the  5  •  of the fecond )  the  re  ft  angle  paralleled 
gramme  contained  lender  the  lines  AE  and  E  C  together  "frith  the  fquare  of  tf?e 
line  E  G  is  equall  to  the fquare  of  the  line  G  C.  Tut  the  fquare  of  the  line  G  F 
common  to  them  both  pfrher fore  that  "frhich  is  contained  Imder  the  lines  AEtsr 
E  C  together  -tyitb  thefquares  of  the  lines  EG  and  G  F  is  equall  to  thefquares  of 
the  lines  GFEr  G  C.  But  l?nto y  fquares  ofy  lines  EG  iy  GF is  equally  fquare 
ofy  line  F  E  (by  the  4-1. of  the frit):  and  to  the Jquares  of the  lines  G  C  andGF 
is  equall  the  'fquare  of  the  line  l  C  (by  the fame)  Wherfore  that  1 frhich  is  contain 
nfdynderthe  lines  A  E  and  E  Cpogether  with  the  fquare  of  the  line  F  E  is  e* 
quail  to  the  fquare  of  the  line  FC.  But  the  line  F  C  is  equall  to  the  line  FB.  For 
they  are  drawen  from  the  centre  to  the  circumference.  Wherfore  that  "frhich  is 
contained  hnder  the  lines  A E  and  E  C  together  1 mb  the fquare  of  the  lyne  FE 
is  equal  to  the  fquare  of the  line  F  B.And  by  the fame  demonftration  that  "frhich 
is  contained  lender  the  lines  ©  E  and  E  B  together  "frith  the fquare  of  the  line  F 
E  is  equall  to  the  fquare  of  the  line  F  B.  Wherfore  that  "frhich  is  contained  "bn* 
der  the  lines  A  E  and  E  C  together  -frith  the  fquare  of  the  line  E  F  is  equal l  to 
that  - frhich  is  contayned  Imder  the  lines  ©  E  and  E  B  together " frith  the  fquare 
of  the  line  EF.T ake  away  the fquare  of  the  line  EE  "frhich  is  common  to  them 
both. Wherfore  the  re  ft angle  par  allelogramme  remayning  -frhich  is  contayned 
Imder  the  lines  A  E  and  E  C  is  equall  to  the  reftangle  par  allelogramme  remay * 
idngj  which  is  contayned  bn  der  the  lines  ©  E  and  E  B.  If therefore  in  a  circle 
two  fight  lines  do  cut  the  one  the  other :  the  reftangle  par  allelogramme  compre* 

Ee.  tj ,  bended 


the  reftangle parallelogramme  contained  frnder  the 
partes  ID  E  and  E  B.For  if  the  line  A C  and  B  ID 


Two  Cafes  its 
this  fiopo/i- 
tion , 

Ftrft  cafe* 
Detnonftra* 

tion. 


The  fecond 

C*f<?r 

tonfruetkn * 


Vemotiflra - 
s'wu 


Three  cafssin 
this  I’ropQj'i- 
than . 


The  third 
safe. 


-i 


7 he  ttiirdTfooke 

hendedlander  the  fegmentes  or  parts  of  the  one  line  is  equall  to  the  reBan gle  pa$ 
rallelograme  comprehended  lender  the  fegmentes  or  parts  of the  other liner^hich 
"'was  required  to  he  demonUrated. 

In  thys  Proportion  are  three  cafes : For  eyther  both  the  lines  paffebytheceiv 
tte,  or  n  eyther  of  them  paiTcth  by  the  centre :  or  the  one  paffeth  by  the  centre  and 
tire  other  not.  The  two  firlb  cafes  are  before.dcmonilrated.  •*- 7" 

But  now  let  one  of  the  lines  onely,  namely,  the  line  zAC  pafle  by  the  centre,  which 
let  be  the  poynt  F,  and  let  it  cut  the  other  line,  namely,  B  E,  in  thepoynt  E  .  Now 
then  the  lind  AC  deuideth  theline  BE  eyther  into  two  equall  partes,,  or  into  two  vn- 
equall  partes .  Pyrft  let  it  deuide  it  into  two  equall  partes  :Whcreforefa!fo  it  deuideth  it 
right  angled  wyfe  by  the  5.  of  thys  booke .  Drawc  aright  line  from  B  to  F.  Where¬ 
fore  B  EF  is  a  right  angled  triangle .  And  for  afmnch  as  the  right  line  AC  is  deuided 
into  two  equal!  partes  in  the  poynt  F,&  into  two  vnequall  partesjin  the  poynt  £ .  Ther- 
fore  the  .redangle  figure  contayned  vnder  the 
lines  zA  E  and  E  C  together  with  the  fquare  of 
the  line  E  f,i  $  equall  to  the  fquare  of  the  line  F  C 
(by  the  5.  of  the  fecond).  But  v.nto  the  fquare  of 
the  line  FC  is  equal!  the  fquare  of  the  line  2?  F 
/for  that  the  lines  F  B  and  F  C are  equall).  Ther- 
fore  that  which  is  cctayned  vnder  the  lines  AE 
and  E  C  together  with  the  Square  of  the  line  E  F, 
is  equall  to  the  fquare  of  the  line  B  F,  Butvnto 
•the Square: of th'ciinc B'F, are  equall  the  fquares 
oftheiints  BEzndEF  (by  the  47.  of  the  hrft). 

Wherefore  that  which  is  contayned  vnder  the 
tines  %AE  and  EC  together  with  the  fquare  of 
theline  E  F.,  is  equall  to  the  fquares  of  the  lines 
ME  and  EF.  lake  away  the  fquare  of  the  line  ,  - 

EF  which  is  common  to  them  both  ;  Wherefore  that  which  remayneth,  namely,  that 
which  is  contayned  vnder  the  lines  *A  £  and  EC,  is  equall  to  the  refidue,  namely,  to 
the  fquare  o  f  t  he  line  BE.  B  ut  the  fq  uare  of  the  line  B  £  is  that  which  is  contained  vn¬ 
der  the  lines  B  £  and  E  E  for  (by  fuppofition)  the  line  BE  is  equall  to  theline  E  E, 
Wherefore  that  which  is  contayned  vnder  the  lines  A  E  &£C,is  equall  to  that  which 
is  contayned  vnder  the  lines  BE  and  E  E  s  which  va$  required  to  beproued. 

But  uowlet  the  line  zA Cpaffing  by  the  centre, 
deuide;  the  line  B  D  notpaffing  by  the  centre, vn- 
equalW in  the  poynt  E .  And  fro  the  poynt  £  raife 
vp  vnto  the  line  &AC  a  perpendicular  line  E  H, 
which  produce  on  the  other  fide  to  the  poypt  G. 

Wperefpre  (by  the  3.  ofthisbooke)  the  line  EH 
is  equall  to  the  line  E  G .  Whertore  as  we  haue  be¬ 
fore  proued ,  that  which  is  contayned  vnder  the  & 
lines  A  E  and  £ C,  is  equall  to  that  which  is  eon- 
tayned  vnder  the  lines  GE  &  E  H ;  but  that  which 
is  contayned  vnder  the  lines  B  E  and  £<Zhisalfo  < 
equall  to  tbft  which  is  contayned  vnder  the  lines 
G  E  and  £  H,  by  the  fecond  cafe  of  thys  Propositi¬ 
on  :  Wherfore  that  which  is  contayned  vnder  the 
lines  e^£  and  £Cris  equall  to  that  which  is  con-  ■  ,v 

tayned  vnder  the  lines  BE  and  EE  1  which  was  agayne  required  t?  be  proued. 

Amongeftali  the  Propofitions  in  this  third  booke3doubtles  thys  is  one  of  the 
chiefeft .  For  it  fetteth  forth  vnto  vs  the  wonderful!  nature  of  a  circle  .  So  that  by 

« ■  7  *•  r.  w  -  it  '7 


-  M 


.  _  ■  \ 

\ _ \ 

F  •  - 

E 

/ 

\  ■ 

c 

m 


v 


ofSuclides  Element  es . 


Foh\ojt 


it  may  be  done  many  goodly  conclufions  in  Geometry  ,  as  (hall  afterward  be  de¬ 
clared  when  occafion  lhall  feme. 

’W  '*  •*  "  *  '*  '  *  '  ' 

£yThef  o. Theorem  e.  The  %6.  Tropojition* 

*  ■  - '  *  ■.  \\  ,  if.  \  -  ..  ,  'V:.  •'  *  ■  .  ..  ..  •-  -■  \  > -•_  /  _• 

If  without  a  circle  he  tahgn  a  certaine  points  and from  that 
point  he  drawen  to  the  circle  two  rift  lines  fo  that  the  one  of 
them  do  cut  the  circle ?  and  the  other  do  touch  the  circle:  the 
rectangle  parallel ogramm  e  which  is  comprehended  "snider  the 
whole  right  line  which  cutteth  the  circle,  and  that  portion  of 
the  fame  line  that  hetb  hetwene  the  point  and  the  vttercir > 
deference  of  the  circle ,  i  s  equall  to  the fquare  made  of  the  line 
that  touched  the  circle. 

*  •'  -  .  .  >  ei  ■%  i  2d.  . 

Vppoje  that  the  circle  be  AB  C :  and  without  the  fame  circle  taken* 
ny  point  at  all  aduentures/nd  let  the fame  he  CD,  And  from  the  point 
!  D  let  there  be  drawen  to  the  circle  two  right  lines  DC  A  and  D 
and  let  the  right  line  DC  A  cut  the  circle  AC  Bin  the  point  C /aid 
let  the  right  line  B  D  touch  the  fame.  'Then  I jay }  that  the  reftangle parallels • 
gramme  contayned  "tinder  the  lines  AD  and  D  C,  is  e  quail  to  the  fquare  of the 
dine  BD .  IsLow  the  line  DC  A  is  either  drawen  by  the  centre /r  not . 

Fir  ft  let  it  be  drawen  by  the  centre .  And  (by 
the  frft  of the  thirdjlet  the  poynt  F  hey  centre  of 
the  circle  ABC }  and  drdwe  a  line  from  F  to  B< 

Wherefore  the  angle  FBD  is  aright  angle.  And 
for  afmuch  as y  right  line  A  C  is  deuided  into  two 
equall partes  in  the  poynt  F/nd  "tinto  it  is  added 
directly  a  right  line  C  D /her  for e( by  the  6 .  of  the 
fgcond  )  that  'Svhicb  is  contayned  "Snider  the  lines 
A  D  and  D  C  together  yoith  the  Jquare  of  y  line 
C  F,  is  e quail  to  the 'Jquare  of  the  line  FD.  But 
the  Ime  F  C  is  e quail  to  the  line  F  B}  for  they  are 
drawen  from  the  centre  toy  circumference:  Wher * 
fore  that  lohich  is  contayned ". tinder  the  lines  AD 
and  D  C  together  t>ith  the  fquare  of  the  line  FB, 
ts  equall  to  the  fquare  of  the  line  FD.Buty  fquare 
of  the  line  FD>  is  (by  the  47.  of  thefirftj  equall 
to  the '/quarts  of the  lines  F  B  and  BD  (for  the 
angle  FBD  is  a  right  an  fee) .  Wherefore  that  which  is  contayned  "tinder  tfa 
lines  AD  and  DC  together  With  the  Jquare  of  the  line  FByis  equall  to  the 

Ee.iij.  j quarts 


CotlftyuakXi 


Two  cafes  in 
this  Propofi - 
tiov. 

‘IbefirRcapte 


DmmUra- 


Tktf?c9nA 

stft, 

Conflmtim ♦ 


VemonffM* 

eion. 


fquares  of  the  lines  F  $  and  T  0 .  Take  away  the fquartof the  lineFTidhkh 
k  common  to  them  both .  Wherefore  thatlohich  remaynethy  namely,  that  lohicb 
is  contayned lender  the  lines  AT)  and  D  Cy  is  equall  to  the  fquare  made  of  the 
line  D  B  lohichtoucheth  the  circle. 

Tut  now  fuppofe  that  the  right  line  D  C^Ahe 
not  drawen  by  the  centre  of  the  circle  ATC.  And 
(by  the  fir  ft  of the  third)  let  thepoint  E  bey  cen <* 
tre  of  the  circle  fifT  C.  And  from)  poynt  Efiraw 
(by  the  12.  of  the  fir  ft)  Tmto  the  line  AC  a  per* 
pendicular line  EE  y  and  draw  thefe  right  lines 
ETjECynd  ET)  .TSLow  the  angle  EE  ID  is  a 
right  angle  .Andforafmuch  as  a  certaine  right 
line  E  E  drawen  by  the  centre yutteth  a  certayne  E 
other  right  line  AC  not  drawen  by  the  centre  fin 
fuch fort  that  it  maketh  right  angles  3  it  deuideth 
it(byy  third  of the  third)  into  two  equall partes. 

Wherefore  the  line  A  F  is  equall  to  the  line  FC. 

And  for afmuch  as  the  right  line  A  C  is  deuided 
into  two  equall  partes  in  the  poynt  Fyi?  7>nto  it  is 
added  directly  an  other  right  line  making  both 
me  right  line y  therefore  (by  the  6 .  of  the fecond) 
thatlohich  is  contayned  Crider  the  lines  Djland  D  C  together ‘With  the  fquare 
of the  line  E  Cy  is  equall  to  the  fquare  of the  line  F  0 ;  put the fquare  of the  line 
E  E  common  to  them  both.  Wherefore  that  nhich  is  contayned  Imder  the  lines 
0  A  and  0  C  together  1 vith  the Jquares  of  the  lines  C  F  andFEy  is  equall  to 
the  fquares  of  the  lines  F 0  and  EE.  Tut  to  the fquares  of  the  lines  F0  and 
F  Ey  is  equall  the  fquare  of the  line  DE(  by  the  4.7 -of  1 the  firfl  )for  the  angle 
EE D  is  a  right  angle .  „ And  to  the fquares  of the  lines  C  F  and  EE,  is  equall 
the fquare  of  the  line  CE  (by  the fame).  Wherfore  that  ahich  is  contayned  Tut* 
der  the  lines  A  0  and  0  C  together  1 nth  the fquare  of the  line  EC  fits  equall  to 
the  fquare  of  the  line  ED .  Tut  the  line  EC  is  equall  to  the  line  ET: for  they 
are  drawen  from  the  centre  to  the  circumference .  Wherefore  that  lohicb  is  con « 
tayned  lendei  the  lines  A  D  and  0  C  together  loith  the  fquare  of the  line  E  Ts 
is  equall  to  the  fquare  of  the  line  ED .  Tut  to  the  fquare  of  the  line  EDyaree* 
quail  the fquares  of the  lines  E  T  and  TD(  by  the  47.  of the firSl)for  the  an* 
gle  ETD  is  a  right  angle :  Wherefore  that  lohicb  is  contayned  lender  the  lines 
A  D  and  DC  together  loith the fquare  of the  line  ETfis  equall  to  the  fquares 
of  the  lines  E  T  and  BD  .Takeaway  the  fquare  of the line ET  lohich  is  com* 
mon  to  them  both:  Wherefore  the  refiduey  namely y  thatlohich  is  contayned 'ion* 
der  the  lines  jiD  and  D  Cy  is  equall  to  the  fquare  of  the  line  D  T .  If  therfore 
loithout  a  circle  be  taken  a  certaine  point  y  and from  that  poynt  be  drawen  to  the 
circle  two  right  lines yfo  that  the  one  of  them  do  cut  the  circle  f  and  the  other  do 


ofSttcliJes  Elemenies. 

touch  Whclrcle  i  We.  reftangk  pdralhlogramme  isMch  is  Comprehended  Tmdef 
the  lehole  right  fine  uhichcutteththe  circle  and  that  portion  of  the  fame  line 
that  Ueth  betwene  the poynt  and  the  latter  circumference  of  the  circle  js  equall  to 
the  fquare  made  of  the  line  that  touche th  the  circle :  lohich  leas  required  to  h 
demonftrated .  * 

%  A |  f  ...  ■_  \]  n  t  ; *!.  ,  1  '  ’•."r  V*  -  •  -  - 

fT  ipo  Corollaries  out  of Campane ■* 

If  from  one  andthe felfe fime  poynt  iakfn  Without  a  circle  be  drawen  into  the  circle  lines  hov» 
many  footer :  the  retiangle  Parallelogf amines  contaynedvnder  every  one  of  them  and  hys  outward 

are  t quail  the  one  to  the  other. 

if .  "  '  4  _ .  •-  ■  _  -  '  • •"  -  ,  ■  ' 

And  thys  is  hereby  manifcfl,  for  that  euciy  one  of  thoie  redangle  Parallelo- 
grammes  arcequali  to  the  fqtrare  of  the  line  which  is  drawen  from  that  poynt  and 
toucheth  the  circle  by  thys  3d* Proportion  .  Hereunto  he  addeth* 

If  two  lines  drawen  from  one  and  the feif:  fame  point  do  touch  a  circle,  they  are  equal!  the  one  to 
the  other.  * 

Which  although  it  neede  no  demcnftration,  for  that  the  fquarc  ofeyther  of 
them  is  equall  to  that  which  is  contayned  vnder  the  line  drawen  from  the  fame 
poynt  ana  hys  outward  part :  yet  he  thus  proueth  it. 

Suppofe  that  there  be  a  circle  B  CD,  whole  A 

centre  let  be  E ,  and  without  it  take  the  point  A „ 

And  from  the  poynt  A  drawe  two  lines  AB  and 
*AD,  which  let  touch  the  circle  in  the  poyntes 
B  and  D .  Then  I  fay , that  they  are  equal! .  Draw* 
thefe  right  lines  EB,ED,  and  AE .  And  by  the 
l8.  of  thys  booke,eyther  of  the  angles  at  the 
poyntes  B  and®  is  a  right  angle.  Wherefore  (by 
the  47.  of  the  frit)  the  fquarc  of  the  line  E, 
is  equall  to  the  two  fquares  of  the  lines  A  B  and 
EB :  and  by  the  fame  reafon,to  the  two  fquares 
ofthelines  AD  and  ED .  Wherefore  the  two 
fquares  of  the  lines  AB  and  EB,axt  equall  to 
the  two  fquares  of  the  lines  tA  D  and  E  D.And 
rforafmuch  as  the  fquares  of  the  lines  EB  and 
ED  are  eqiiallrthereforethetwo  other  fquares 
ofthelines  AB  and  AD  are  alfo  equall  .Wher- 
fore  the  line  AB  is  equall  to  the  line  A  D ; which 
was  required  to  be  proued. 

The  fame  may  be  proued  an  other  way :  Draw  a  line  from  B  to  D,  And  (by  the  $  .61 
the  firft)  the  angle  EB  D  i  s  equall  to  the  angle  E  D  B .  Andforafmuchasthe  two  an¬ 
gles  A B  E  and  <tA  D  E  are  equall,naniely,for  that  they  are  right  angles  :  if  you  take 
from  them  the  equall  angles  EB  D  &  ED  B,  the  two  other  angles  remayning,  namely, 
the  angles  ABD  and  ADB  flialTbe  equall .  Wherefore(  by  the  <5.  of  the  firft}  the  line 
AB  is  equall  to  the  line  A  D. 


\ 


/\ 


\ 

I  • 

E 

)  1 

/  ! 

/  . 

y 


f  Hereunto  alfo  Telitarius  addeth  this  Corollary. 

Trim  a  poynt  geuen  Without  a  circle, can  be  drawen  vnto  a  circle  onelytivo  touch  lines . 

The  former  defeription  remayning, Ifay  that  from  the  poynt  A  can  be  drawen  vnto 

Ee.iiif  the 


Fits!  CortHa- 


Second  Co* 
tddttry. 


Third  Cord* 

lary,  • 


thecircle  BCD  no  more  touch  lines,but  the  tsvo  lines’  A B  and  AD.  Forifiebepof- 
iible,  let  A  F  alfo  be  in  the  former  iigurc  a  touch  line,touching.the  circle  in  the  poynt 
F.Andprawe  a  line  from  E  to  F.  And  the  angle  at  the  point  F  fhall  be  aright  angle,  by 
the  1 8 .  of  this  booke :  Wherefore  it  is  equail  to  the  angle  E  B  A>  which  is  contrary  to 
the  20. of  the  firir. 

This  may  alfo  be  thus  proued.  For  afmuch  as  all  the  lines  drawen  from  one  and-the 
felfe  fame  poynt  &  touching  a  circle  are  equail, as  we  haue  before  proued,but  the  lines 
3  and  AF  can  not  be  equail,  by  the  8.  Propofition  of  this  booke,  therefore  the  line 
*A F  can  not  touch  tfe circle  BCD „ 


y&ffhe  iiSIheoreme*  1 he  37.  Tropofition* 

If  without  a  circle  be  taken  a  cert  nine point ,  and from  that 
point  be  drawen  to  the  circle  two  right  lines  jf  which y the  one; 

■  doth  cut  the  circle  and  the  other  falleth  vpon  the  circle ,  and 
i ha t  in fuch  fort }t hat  the  rectangle parallelogramme  which  is 
cotayned 'under  the  whole  right  line  which  cutteth  the  circle s 
and  that  portion  of  the  fame  line  that  heth  betwene  the  point 
and  the  y tiercircumferece  of  the  circle  fits  equail  to  the  (quart 
made  of  the  line  that  falleth  ypon  the  circle  :  then  that  line 
that  jo  falleth  vpon  the  circle  fhall  touch  the  circle. 


"This  propor¬ 
tion  ts  the  co¬ 
ney/ e  ofibe 
former. 


Confrufmn, 


Vemonftra- 


r Et  the  circle  be  ABC:  andlcith* 

Of  out  the fame  circle  take  a  point }  and 
»  the fame  be  ID, &  from  the  point 

fffiffD  let  there  be  drawen  to  the  circle 
ABC  two  right  lines  ID  C A  and  D B  2  and 
let  DC  A  cut  the  circle }  and  D  B fall  ypon  the 
circle,  And  that  in  juch Jort}  that  that  fhich 
is  contained  lender  the  lines  AD  and  DCy 
be  equail  to  the  fquare  of the  line  DB .  Then 
I fayjhaty  line  D  B  toucheth  the  circle  ABC, 

Drawe  (by  the  17*  of the  third ) from  the  poynt 
D  a  right  line  touching  the  circle  A  B  C  }  and 
let  the  fame  be  DE.  And(  by  the  fir/1  of  the 
Janie )  let  the  point  F  be  the  centre  of the  circle 
ABC:  and  draw  thefe  right  lines  FE,FB, 
and  F D .  Wherfore  the  angle  FE  D  is  a  right 
angle .  And  for  afmu